1 Introduction

This is the analysis using Bayesian Modeling.

2 Data preparation

2.1 Source setup

########## folders ##########
# current folder (first go to session -> set working directory -> to source file location)
parentfolder <- dirname(getwd())

data          <- paste0(parentfolder, '/MultIS_data/')
audiodata     <- paste0(parentfolder, '/audio_processed/')
syllables     <- paste0(audiodata,    'syllables/')
dataworkspace <- paste0(parentfolder, '/data_processed/')
datamerged    <- paste0(parentfolder, '/data_merged/')
datasets      <- paste0(parentfolder, '/datasets/')
models        <- paste0(parentfolder, '/models/')
plots         <- paste0(parentfolder, '/plots/')
scripts       <- paste0(parentfolder, '/scripts/')

########## source file ##########

#source(paste0(scripts, "adjectives-preparation.R"))

#################### packages ####################
# Data Manipulation
library(tibble)
library(stringr)
library(tidyverse) # includes readr, tidyr, dplyr, ggplot2
packageVersion("tidyverse")
## [1] '2.0.0'
library(data.table)

# Plotting
library(ggforce)
library(ggpubr)
library(gridExtra)
library(corrplot)

# Bayesian
library(brms); packageVersion("brms")
## [1] '2.22.0'
library(cmdstanr); packageVersion("cmdstanr")
## [1] '0.8.1'
library(emmeans); packageVersion("emmeans")
## [1] '1.10.5'
library(posterior); packageVersion("posterior")
## [1] '1.6.0'
# tidybayes https://mjskay.github.io/tidybayes/articles/tidy-brms.html
library(magrittr)
library(dplyr)
library(purrr)
library(forcats)
library(tidyr)
library(modelr)
library(ggdist)
library(tidybayes)
library(ggplot2)
library(cowplot)
library(rstan)
library(brms)
library(ggrepel)
library(RColorBrewer)
library(gganimate)
library(posterior)
library(distributional)

theme_set(theme_tidybayes() + panel_border())


# use all available cores for parallel computing
options(mc.cores = parallel::detectCores())

colorBlindBlack8  <- c("#000000", "#E69F00", "#56B4E9", "#009E73", 
                       "#F0E442", "#0072B2", "#D55E00", "#CC79A7")

2.2 Functions

# Function to back-transform fixed effects from a model
backtransform_fixef <- function(model) {
  
  # Extract fixed effects (population-level)
  fixef_output <- fixef(model)
  
  # Extract the estimates for the intercept and percProm levels
  intercept <- fixef_output["Intercept", "Estimate"]
  percProm1 <- fixef_output["percProm1", "Estimate"]
  percProm2 <- fixef_output["percProm2", "Estimate"]
  percProm3 <- fixef_output["percProm3", "Estimate"]
  
  # Combine the effects for each percProm level before applying exp()
  combined_effects <- data.frame(
    percProm_level = c("percProm0", "percProm1", "percProm2", "percProm3"),
    Estimate = exp(c(intercept,  # percProm0 is just the intercept
                     intercept + percProm1,  # Add percProm1 effect to intercept
                     intercept + percProm2,  # Add percProm2 effect to intercept
                     intercept + percProm3)),  # Add percProm3 effect to intercept
    # You can calculate confidence intervals by combining Q2.5 and Q97.5 as well
    Q2.5 = exp(c(fixef_output["Intercept", "Q2.5"],  # percProm0 interval
                 fixef_output["Intercept", "Q2.5"] + fixef_output["percProm1", "Q2.5"],  # percProm1 interval
                 fixef_output["Intercept", "Q2.5"] + fixef_output["percProm2", "Q2.5"],  # percProm2 interval
                 fixef_output["Intercept", "Q2.5"] + fixef_output["percProm3", "Q2.5"])),  # percProm3 interval
    Q97.5 = exp(c(fixef_output["Intercept", "Q97.5"],
                  fixef_output["Intercept", "Q97.5"] + fixef_output["percProm1", "Q97.5"],
                  fixef_output["Intercept", "Q97.5"] + fixef_output["percProm2", "Q97.5"],
                  fixef_output["Intercept", "Q97.5"] + fixef_output["percProm3", "Q97.5"]))
  )
  
  return(combined_effects)
}

combine_fixed_random_backtransforming <- function(participant_id) {
  
  # Get the random effects for the given participant
  rand_intercept <- ranef_output$participant[participant_id, , "Intercept"]
  rand_percProm1 <- ranef_output$participant[participant_id, , "percProm1"]
  rand_percProm2 <- ranef_output$participant[participant_id, , "percProm2"]
  rand_percProm3 <- ranef_output$participant[participant_id, , "percProm3"]
  
  # Combine fixed and random effects
  participant_effects <- data.frame(
    percProm_level = c("percProm0", "percProm1", "percProm2", "percProm3"),
    Estimate = exp(c(
      intercept + rand_intercept["Estimate"], 
      intercept + percProm0 + rand_intercept["Estimate"] + rand_percProm1["Estimate"],
      intercept + percProm2 + rand_intercept["Estimate"] + rand_percProm2["Estimate"],
      intercept + percProm3 + rand_intercept["Estimate"] + rand_percProm3["Estimate"]
    )),
    Est.Error = c(
      sqrt(fixef_output["Intercept", "Est.Error"]^2 + rand_intercept["Est.Error"]^2),
      sqrt(
        fixef_output["Intercept", "Est.Error"]^2 + fixef_output["percProm1", "Est.Error"]^2 +
        rand_intercept["Est.Error"]^2 + rand_percProm1["Est.Error"]^2
      ),
      sqrt(
        fixef_output["Intercept", "Est.Error"]^2 + fixef_output["percProm2", "Est.Error"]^2 +
        rand_intercept["Est.Error"]^2 + rand_percProm2["Est.Error"]^2
      ),
      sqrt(
        fixef_output["Intercept", "Est.Error"]^2 + fixef_output["percProm3", "Est.Error"]^2 +
        rand_intercept["Est.Error"]^2 + rand_percProm3["Est.Error"]^2
      )
    ),
    Q2.5 = exp(c(
      intercept + rand_intercept["Q2.5"], 
      intercept + percProm0 + rand_intercept["Q2.5"] + rand_percProm1["Q2.5"],
      intercept + percProm2 + rand_intercept["Q2.5"] + rand_percProm2["Q2.5"],
      intercept + percProm3 + rand_intercept["Q2.5"] + rand_percProm3["Q2.5"]
    )),
    Q97.5 = exp(c(
      intercept + rand_intercept["Q97.5"], 
      intercept + percProm0 + rand_intercept["Q97.5"] + rand_percProm1["Q97.5"],
      intercept + percProm2 + rand_intercept["Q97.5"] + rand_percProm2["Q97.5"],
      intercept + percProm3 + rand_intercept["Q97.5"] + rand_percProm3["Q97.5"]
    ))
  )
  
  participant_effects$participant <- participant_id
  return(participant_effects)
}

combine_fixed_random_student <- function(participant_id) {
  
  # Get the random effects for the given participant
  rand_intercept <- ranef_output$participant[participant_id, , "Intercept"]
  rand_percProm1 <- ranef_output$participant[participant_id, , "percProm1"]
  rand_percProm2 <- ranef_output$participant[participant_id, , "percProm2"]
  rand_percProm3 <- ranef_output$participant[participant_id, , "percProm3"]
  
  # Combine fixed and random effects for each percProm level
  participant_effects <- data.frame(
    percProm_level = c("percProm0", "percProm1", "percProm2", "percProm3"),
    
    # Estimate: sum of fixed and random effects
    Estimate = c(
      rand_intercept["Estimate"], 
      rand_intercept["Estimate"] + rand_percProm1["Estimate"],
      rand_intercept["Estimate"] + rand_percProm2["Estimate"],
      rand_intercept["Estimate"] + rand_percProm3["Estimate"]
    ),
    
    # Standard Error: combining errors using sqrt of sum of squares
    Est.Error = c(
      sqrt(rand_intercept["Est.Error"]^2),
      sqrt(rand_intercept["Est.Error"]^2 + rand_percProm1["Est.Error"]^2),
      sqrt(rand_intercept["Est.Error"]^2 + rand_percProm2["Est.Error"]^2),
      sqrt(rand_intercept["Est.Error"]^2 + rand_percProm3["Est.Error"]^2)
    ),
    
    # Q2.5 and Q97.5: these need to be simulated to reflect the full posterior distribution
    Q2.5 = c(
      rand_intercept["Q2.5"], 
      rand_intercept["Q2.5"] + rand_percProm1["Q2.5"],
      rand_intercept["Q2.5"] + rand_percProm2["Q2.5"],
      rand_intercept["Q2.5"] + rand_percProm3["Q2.5"]
    ),
    Q97.5 = c(
      rand_intercept["Q97.5"], 
      rand_intercept["Q97.5"] + rand_percProm1["Q97.5"],
      rand_intercept["Q97.5"] + rand_percProm2["Q97.5"],
      rand_intercept["Q97.5"] + rand_percProm3["Q97.5"]
    )
  )
  
  participant_effects$participant <- participant_id
  return(participant_effects)
}

# Function to compute summary statistics for each difference
compute_summary <- function(diff) {
  estimate <- mean(diff)
  se <- sd(diff)
  ci_lower <- quantile(diff, 0.025)
  ci_upper <- quantile(diff, 0.975)
  prob <- mean(diff > 0)  # Posterior probability that difference > 0
  return(data.frame(Estimate = estimate, SE = se, CI.Lower = ci_lower, CI.Upper = ci_upper, Prob = prob))
}

2.3 Load in data frames

participant_info <- read_delim(paste0(data,"ParticipantInfo_GERCAT.csv"), delim = ";")

# Load the information about duration of each segment (if needed)
data_df <- read.table(paste0(syllables, "fileDurationsDF.csv"), header = TRUE, sep = ',')

# Load cleaned syllable data
data <- read_csv(paste0(datasets, "data_cleaned.csv"))

# Load cleaned targets data
targets <- read_csv(paste0(datasets, "targets.csv"))

# Load cleaned targets with pre-post data
data_prepost <- read_csv(paste0(datasets, "data_prepost.csv"))

2.4 You can add participant info

# Process participant_info so that participant number column is only number
participant_info$Participant <- parse_number(participant_info$Participant)

# Convert the column names of participant_info to lowercase
colnames(participant_info) <- tolower(colnames(participant_info))

# Merge the dataframes by "Participant" and "Language"
data_prepost <- merge(data_prepost, participant_info, by = c("participant", "language"), all.x = TRUE)

3 Data preparation

We split the table in two languages. Then, we only keep the relevant columns (top 10 features).

# Turn variables to factors
data_prepost$percProm <- as.factor(data_prepost$percProm)
data_prepost$itemNum <- as.factor(data_prepost$itemNum)
data_prepost$focus <- as.factor(data_prepost$focus)
data_prepost$participant <- as.factor(data_prepost$participant)

# Add contrast-coded gender info (in case we want to use if)
data_prepost <- data_prepost %>% 
  mutate(gender_s = case_when(gender == "female" ~ 0.5,
                              gender == "male" ~ -0.5))

# First, remove some columns for both languages
data_prepost <- data_prepost %>%
  select(-f1_freq_median, -f1_freq_median_norm, -f2_freq_median, -f2_freq_median_norm, 
         -f1_freq_medianPre, -f1_freq_median_normPre, -f2_freq_medianPre, -f2_freq_median_normPre, 
         -f1_freq_medianPost, -f1_freq_median_normPost, -f2_freq_medianPost, -f2_freq_median_normPost
         )

# Adapt duration scale (because we set the priors for ms, not s)
data_prepost$duration <- data_prepost$duration * 1000
data_prepost$durationPre <- data_prepost$durationPre * 1000
data_prepost$durationPost <- data_prepost$durationPost * 1000
data_prepost$duration_noSilence <- data_prepost$duration_noSilence * 1000
data_prepost$duration_noSilencePre <- data_prepost$duration_noSilencePre * 1000
data_prepost$duration_noSilencePost <- data_prepost$duration_noSilencePost * 1000

# Create data_prepost_german for rows where language is German
data_prepost_ger <- data_prepost %>% 
  filter(language == "German")

# Create data_prepost_catalan for rows where language is Catalan
data_prepost_cat <- data_prepost %>% 
  filter(language == "Catalan")

# Select columns for German
data_prepost_ger <- data_prepost_ger %>%
  select(fileName, language, participant, gender, gender_s, age, 
         itemNum, focus, annotationNum, annotationNumTarget, 
         word, syllText, syllTextPre, syllTextPost, percProm,
         f0_slope, f0_slope_norm,
         pitch_median, pitch_median_norm,
         ampl_sd,
         ampl_noSilence_medianPost,
         ampl_noSilence_median,
         duration,
         flux_sd,
         flux_median,
         ampl_noSilence_sd,
         pitch_medianPost, pitch_median_normPost)

# Select columns for Catalan
data_prepost_cat <- data_prepost_cat %>%
  select(fileName, language, participant, gender, gender_s, age, 
         itemNum, focus, annotationNum, annotationNumTarget, 
         word, syllText, syllTextPre, syllTextPost, percProm,
         duration,
         f0_slopePost, f0_slope_normPost,
         fmDep_medianPost,
         entropySh_sd,
         flux_medianPost,
         ampl_medianPost,
         specCentroid_median,
         duration_noSilence,
         durationPre,
         ampl_sd)

Add information on pauses to test separately. First in German.

# First, let's create a copy of data_prepost_ger to work on
data_prepost_ger_updated <- data_prepost_ger

# Add a new column durationPause and pausePre initialized to NA
data_prepost_ger_updated$durationPause <- NA
data_prepost_ger_updated$pausePre <- NA

# Loop through each row in data_prepost_ger where syllTextPre is NA
for (i in 1:nrow(data_prepost_ger_updated)) {
  # Check if syllTextPre is NA
  if (is.na(data_prepost_ger_updated$syllTextPre[i])) {
    
    # Get the corresponding fileName in this row
    fileName_i <- data_prepost_ger_updated$fileName[i]
    
    # Find the index of this fileName in the data dataset
    match_idx <- which(data$fileName == fileName_i)
    
    # If a match is found, and the previous row exists (to avoid out of bounds)
    if (length(match_idx) > 0 && match_idx > 1) {
      # Look at the previous row's syllText to see if it's "-p-"
      if (data$syllText[match_idx - 1] == "-p-") {
        # Extract the duration value from the previous row
        duration_value <- data$duration[match_idx - 1]
        
        # Update syllTextPre with "-p-" in data_prepost_ger
        data_prepost_ger_updated$syllTextPre[i] <- "-p-"
        
        # Update the durationPause column with the value from data
        data_prepost_ger_updated$durationPause[i] <- duration_value
      }
    }
  }
}

rm(i, fileName_i, duration_value, match_idx)

# Create the pausePre column based on whether durationPause is NA or not
data_prepost_ger_updated$pausePre <- ifelse(is.na(data_prepost_ger_updated$durationPause), "no", "yes")

data_prepost_ger <- data_prepost_ger_updated
rm(data_prepost_ger_updated)

Inspect.

table(data_prepost_ger$pausePre)
## 
##   no  yes 
## 2341    3
prop.table(table(data_prepost_ger$pausePre))
## 
##          no         yes 
## 0.998720137 0.001279863

It does not really make sense to investigate this in German.

Now in Catalan.

# First, let's create a copy of data_prepost_cat to work on
data_prepost_cat_updated <- data_prepost_cat

# Add a new column durationPause and pausePre initialized to NA
data_prepost_cat_updated$durationPause <- NA
data_prepost_cat_updated$pausePre <- NA

# Loop through each row in data_prepost_cat where syllTextPre is NA
for (i in 1:nrow(data_prepost_cat_updated)) {
  # Check if syllTextPre is NA
  if (is.na(data_prepost_cat_updated$syllTextPre[i])) {
    
    # Get the corresponding fileName in this row
    fileName_i <- data_prepost_cat_updated$fileName[i]
    
    # Find the index of this fileName in the data dataset
    match_idx <- which(data$fileName == fileName_i)
    
    # If a match is found, and the previous row exists (to avoid out of bounds)
    if (length(match_idx) > 0 && match_idx > 1) {
      # Look at the previous row's syllText to see if it's "-p-"
      if (data$syllText[match_idx - 1] == "-p-") {
        # Extract the duration value from the previous row
        duration_value <- data$duration[match_idx - 1]
        
        # Update syllTextPre with "-p-" in data_prepost_cat
        data_prepost_cat_updated$syllTextPre[i] <- "-p-"
        
        # Update the durationPause column with the value from data
        data_prepost_cat_updated$durationPause[i] <- duration_value
      }
    }
  }
}

rm(i, fileName_i, duration_value, match_idx)

# Create the pausePre column based on whether durationPause is NA or not
data_prepost_cat_updated$pausePre <- ifelse(is.na(data_prepost_cat_updated$durationPause), "no", "yes")

data_prepost_cat <- data_prepost_cat_updated
rm(data_prepost_cat_updated)

What are the values in Catalan?

table(data_prepost_cat$pausePre)
## 
##   no  yes 
## 2148   98
prop.table(table(data_prepost_cat$pausePre))
## 
##         no        yes 
## 0.95636687 0.04363313

It makes more sense.

4 Modeling

Given the goal of comparing the thresholds of changes in acoustic features across prominence levels, while accounting for speaker variability differences, a hierarchical (or mixed-effects) Bayesian model with acoustic features as outcome variables will be a suitable approach. This model allows you to account for the variability across participants while focusing on estimating the thresholds.

4.1 Catalan

Let’s check the correlations between acoustic features.

# Calculate the correlation matrix
corrCat <- cor(data_prepost_cat[, c("duration", "f0_slopePost", "fmDep_medianPost", 
                                    "entropySh_sd", "flux_medianPost", 
                                    "ampl_medianPost", "specCentroid_median", "duration_noSilence", 
                                    "durationPre", "ampl_sd")], use = "complete.obs")
corrplot(corrCat, method = "color", tl.cex = 0.7, number.cex = 0.7, addCoef.col = "black")

Let’s set contrasts for comparisons.

# Drop unused levels of percProm
data_prepost_cat$percProm <- droplevels(data_prepost_cat$percProm)
# Set sum contrasts for percProm to compare all levels
#contrasts(data_prepost_cat$percProm) <- contr.sum(length(unique(data_prepost_cat$percProm)))

What is the mean for gender.

summary(data_prepost_cat$gender_s)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -0.5000 -0.5000  0.5000  0.1563  0.5000  0.5000
# 0.1563

We go one by one the ten features from top to bottom.

4.1.1 Cumulative

data_prepost_cat$percProm <- factor(data_prepost_cat$percProm, 
                                   levels = c(1, 2, 3), 
                                   ordered = TRUE)

4.1.1.1 Formulas

durCum <- bf(
  percProm ~ 
     duration + 
     f0_slopePost + 
     fmDep_medianPost +
     entropySh_sd +
     flux_medianPost +
     ampl_medianPost +
     specCentroid_median +
     durationPre +
     ampl_sd +
    (1 | participant)
)

durCum_ranSlopes <- bf(
  percProm ~ 
     duration + 
     f0_slopePost + 
     fmDep_medianPost +
     entropySh_sd +
     flux_medianPost +
     ampl_medianPost +
     specCentroid_median +
     durationPre +
     ampl_sd +
    (1 + duration + 
     f0_slopePost + 
     fmDep_medianPost +
     flux_medianPost +
     ampl_medianPost +
     specCentroid_median +
     durationPre +
     ampl_sd | participant)
)

durCum_allInt <- bf(
  percProm ~ 
    (duration + 
     f0_slopePost + 
     fmDep_medianPost +
     entropySh_sd +
     flux_medianPost +
     ampl_medianPost +
     specCentroid_median +
     durationPre +
     ampl_sd) * 
    (duration + 
     f0_slopePost + 
     fmDep_medianPost +
     flux_medianPost +
     ampl_medianPost +
     specCentroid_median +
     durationPre +
     ampl_sd) +
    (1 + duration + 
     f0_slopePost + 
     fmDep_medianPost +
     flux_medianPost +
     ampl_medianPost +
     specCentroid_median +
     durationPre +
     ampl_sd | participant)
)

4.1.1.2 Priors

4.1.1.3 Model

pp_check(mdl_durCumCat)

conditional_effects(mdl_durCumCat, re_formula = NA, categorical = TRUE)

hypothesis(mdl_durCumCat, c("duration = 0",
                            "f0_slopePost = 0",
                            "fmDep_medianPost = 0",
                            "entropySh_sd = 0",
                            "flux_medianPost = 0",
                            "ampl_medianPost = 0",
                            "specCentroid_median = 0",
                            "durationPre = 0",
                            "ampl_sd = 0"))
## Hypothesis Tests for class b:
##                 Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio
## 1           (duration) = 0     0.01      0.00     0.01     0.01         NA
## 2       (f0_slopePost) = 0    -0.66      0.18    -1.02    -0.30         NA
## 3   (fmDep_medianPost) = 0    -0.01      0.11    -0.22     0.20         NA
## 4       (entropySh_sd) = 0     1.23      1.65    -2.01     4.50         NA
## 5    (flux_medianPost) = 0     3.35      1.69     0.01     6.66         NA
## 6    (ampl_medianPost) = 0     1.65      0.91    -0.14     3.43         NA
## 7 (specCentroid_med... = 0     0.00      0.00    -0.01     0.00         NA
## 8        (durationPre) = 0     0.00      0.00     0.00     0.01         NA
## 9            (ampl_sd) = 0     1.92      1.60    -1.21     5.10         NA
##   Post.Prob Star
## 1        NA    *
## 2        NA    *
## 3        NA     
## 4        NA     
## 5        NA    *
## 6        NA     
## 7        NA    *
## 8        NA    *
## 9        NA     
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.

4.1.2 Duration

4.1.2.1 Priors

4.1.2.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.1724661 0.01381479 0.1460578 0.1997325

An R² of approximately 0.17 suggests a modest fit. This value indicates that the model explains about 17.2% of the variance in the dependent variable duration. In other words, a small portion of the variability in duration is accounted for by the predictors in the model. The 95% credible interval, ranging from 14.6% to 19.9%, shows some uncertainty around this estimate but provides a stable indication of the model’s limited explanatory power.

4.1.2.3 Fixed effects

Let’s check how it looks like.

# Summarize the model
summary(mdl_durationCat)
##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: duration ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_cat (Number of observations: 2246) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1)                0.16      0.03     0.11     0.23 1.00     7165
## sd(percProm2)                0.14      0.02     0.11     0.18 1.00     6430
## sd(percProm3)                0.18      0.04     0.12     0.27 1.00     7623
## cor(percProm1,percProm2)     0.66      0.15     0.32     0.90 1.00     4310
## cor(percProm1,percProm3)     0.47      0.23    -0.02     0.85 1.00     4645
## cor(percProm2,percProm3)     0.68      0.15     0.32     0.92 1.00     6921
##                          Tail_ESS
## sd(percProm1)                9788
## sd(percProm2)                9675
## sd(percProm3)                9345
## cor(percProm1,percProm2)     6261
## cor(percProm1,percProm3)     7693
## cor(percProm2,percProm3)     9172
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1     5.11      0.03     5.04     5.18 1.00     6387     8711
## percProm2     5.22      0.03     5.17     5.27 1.00     6428     9582
## percProm3     5.35      0.04     5.27     5.43 1.00     8611    11633
## gender_s      0.06      0.04     0.00     0.15 1.00     6584     8062
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.33      0.01     0.32     0.34 1.00    23416    11411
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
# Extract fixed effects
exp(fixef(mdl_durationCat))
##             Estimate Est.Error       Q2.5      Q97.5
## percProm1 165.351091  1.034530 154.740478 176.985674
## percProm2 184.973169  1.026270 175.846273 194.738125
## percProm3 211.451562  1.041156 194.961916 228.371297
## gender_s    1.064828  1.041079   1.003856   1.166572

It shows a stable increase from 1 to 3.

Diagnostic plots.

plot(mdl_durationCat)

Posterior predictive check.

pp_check(mdl_durationCat, ndraws = 100)

pp_check(mdl_durationCat, type = "error_scatter_avg", ndraws = 100)

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##      56     144     180     201     238     709
4.1.2.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  1           165       155       177
##  2           185       176       195
##  3           212       195       228
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm1 - percProm2    -19.6     -29.7     -9.47
##  percProm1 - percProm3    -46.2     -62.9    -29.25
##  percProm2 - percProm3    -26.6     -40.5    -12.20
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.1.2.3.2 Hypothesis testing
  • diff_1_2: Estimate = -19.6 [-29.6, -9.36], Posterior Probability = 99.9%; Strong evidence suggests that the duration at percProm2 is higher than at percProm1. The credible interval does not include zero, and the high posterior probability confirms this difference.

  • diff_1_3: Estimate = -46.2 [-62.9, -29.3], Posterior Probability = 100%; Strong evidence indicates that the duration at percProm3 is higher than at percProm1. The negative estimate and credible interval entirely below zero show a reliable difference between these levels.

  • diff_2_3: Estimate = -26.6 [-40.8, -12.5], Posterior Probability = 100%; Strong evidence that duration at percProm3 is higher than at percProm2. The credible interval is entirely negative, supporting this increase in duration with rising percProm.

Overall Implications:

  • Trend: The results indicate a consistent increase in duration across percProm levels, moving from percProm1 to percProm3.

  • Interpretation: Each higher level of percProm is associated with a reliably longer duration. The strong posterior probabilities and credible intervals suggest that as perceived prominence (percProm) increases, so does duration, supporting a stable trend in the data.

4.1.2.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.1.2.4 Group-level effects

First, get a general overview.

  • percProm1: The estimated standard deviation is 0.160 (95% CrI: 0.108 to 0.225), indicating moderate variability in participants’ response to percProm1.

  • percProm2: The estimated standard deviation is 0.138 (95% CrI: 0.105 to 0.182), suggesting slightly less variability in participants’ response to percProm2 compared to percProm1.

  • percProm3: The estimated standard deviation is 0.184 (95% CrI: 0.119 to 0.266), showing increased variability in response to percProm3 relative to the other prominence levels, indicating that participants exhibit a wider range of responses at this level.

Overall Interpretation: These results suggest that as perceived prominence increases from percProm1 to percProm3, the variability in participants’ responses tends to fluctuate, with percProm3 showing the highest variability. This pattern may imply that the highest level of perceived prominence elicits a more diverse range of responses in terms of durationCat.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.1.3 F0 slope (post-tonic)

4.1.3.1 Priors

4.1.3.2 Model

Calculate R².

##      Estimate   Est.Error       Q2.5      Q97.5
## R2 0.06645816 0.007244046 0.05251865 0.08075153

An R² of approximately 0.07 suggests a weak fit, indicating that the model explains only about 6.6% of the variance in the dependent variable f0_slopePostCat. In other words, a small portion of the variability in f0_slopePostCat is accounted for by the predictors in the model. The 95% credible interval, ranging from 5.3% to 8.1%, reflects some uncertainty around this estimate, but it generally confirms the model’s limited explanatory power.

4.1.3.3 Fixed effects

Let’s check how it looks like.

# Summarize the model
summary(mdl_f0_slopePostCat)
##  Family: student 
##   Links: mu = identity; sigma = identity; nu = identity 
## Formula: f0_slopePost ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_cat (Number of observations: 1780) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1)                0.03      0.02     0.00     0.06 1.00     3199
## sd(percProm2)                0.08      0.01     0.06     0.11 1.00     6465
## sd(percProm3)                0.09      0.02     0.05     0.14 1.00     6050
## cor(percProm1,percProm2)     0.15      0.41    -0.73     0.85 1.00      691
## cor(percProm1,percProm3)     0.16      0.43    -0.72     0.88 1.00     1503
## cor(percProm2,percProm3)     0.36      0.24    -0.18     0.76 1.00    10503
##                          Tail_ESS
## sd(percProm1)                6030
## sd(percProm2)                9786
## sd(percProm3)                6791
## cor(percProm1,percProm2)     1116
## cor(percProm1,percProm3)     2586
## cor(percProm2,percProm3)    11607
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1     0.00      0.01    -0.02     0.03 1.00     7434    10709
## percProm2     0.06      0.02     0.03     0.09 1.00     7191     9945
## percProm3    -0.08      0.02    -0.12    -0.03 1.00    10715    11007
## gender_s     -0.01      0.02    -0.05     0.03 1.00     9400     9275
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.13      0.01     0.12     0.14 1.00    12273    12072
## nu        2.00      0.14     1.75     2.29 1.00    14263    11993
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
# Extract fixed effects
fixef(mdl_f0_slopePostCat)
##               Estimate  Est.Error        Q2.5       Q97.5
## percProm1  0.004748638 0.01197570 -0.01867902  0.02921571
## percProm2  0.062793941 0.01593530  0.03176724  0.09456290
## percProm3 -0.078127142 0.02124579 -0.11879674 -0.03435458
## gender_s  -0.010273064 0.01895841 -0.04863810  0.02640334

It shows a stable increase from 1 to 2, and then a drop from 2 to 3.

Diagnostic plots.

plot(mdl_f0_slopePostCat)

Posterior predictive check.

pp_check(mdl_f0_slopePostCat, ndraws = 100)

pp_check(mdl_f0_slopePostCat, type = "error_scatter_avg", ndraws = 100)

Extract samples.

# Extract posterior samples of fixed effects
samples_mdl_f0_slopePostCat <- as_draws_array(mdl_f0_slopePostCat, 
                                          variable = "^b_", 
                                          regex = TRUE)

# Summarize posterior draws
summarize_draws(samples_mdl_f0_slopePostCat)

Check conditional effect.

# Plot conditional effects (fixed effects only)
conditional_effects(mdl_f0_slopePostCat, re_formula = NA)

# Extract conditional effects data
conditional_effects(mdl_f0_slopePostCat, plot = FALSE, re_formula = NA)$percProm

Compare conditional effects to raw values.

# Calculate raw means by percProm level
data_prepost_cat %>% 
  group_by(percProm) %>%
  summarize(avg_f0_slopePost = mean(f0_slopePost, na.rm = TRUE))
# Summary of f0_slopePost
summary(data_prepost_cat$f0_slopePost)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
## -4.0920 -0.0858  0.0052  0.0324  0.1774  4.2460     466
4.1.3.3.1 Effect comparison
# Compute estimated marginal means on log-scale
em_mdl_f0_slopePostCat <- emmeans(mdl_f0_slopePostCat, ~ percProm)
print(em_mdl_f0_slopePostCat)
##  percProm   emmean lower.HPD upper.HPD
##  1         0.00453   -0.0188    0.0291
##  2         0.06272    0.0315    0.0942
##  3        -0.07870   -0.1201   -0.0358
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95
plot(em_mdl_f0_slopePostCat)

# Perform pairwise comparisons between levels of percProm
emPairs_mdl_f0_slopePostCat <- pairs(emmeans(mdl_f0_slopePostCat, ~ percProm))
print(emPairs_mdl_f0_slopePostCat)
##  contrast              estimate lower.HPD upper.HPD
##  percProm1 - percProm2  -0.0578   -0.0949   -0.0209
##  percProm1 - percProm3   0.0832    0.0383    0.1305
##  percProm2 - percProm3   0.1412    0.0952    0.1861
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95
plot(emPairs_mdl_f0_slopePostCat)

If the comparison includes 0, the contrast is not reliably different.

4.1.3.3.2 Hypothesis testing
# Extract posterior samples of the fixed effects coefficients
as_draws_df(mdl_f0_slopePostCat) %>%
  # Select fixed effects for percProm levels
  select(starts_with("b_percProm")) %>%
  # Rename columns to match percProm levels
  rename_with(~ str_replace(.x, "b_percProm", "percProm"), starts_with("b_percProm")) %>%
  # Compute differences between percProm levels
  mutate(
    diff_1_2 = percProm1 - percProm2,
    diff_1_3 = percProm1 - percProm3,
    diff_2_3 = percProm2 - percProm3
  ) %>%
  # Gather differences into long format
  pivot_longer(
    cols = starts_with("diff_"),
    names_to = "Comparison",
    values_to = "Difference"
  ) %>%
  # Group by comparison to compute summary statistics
  group_by(Comparison) %>%
  summarise(
    Estimate = mean(Difference),
    Est.Error = sd(Difference),
    CI.Lower = quantile(Difference, 0.025),
    CI.Upper = quantile(Difference, 0.975),
    Post.Prob = if_else(Estimate > 0,
                        mean(Difference > 0) * 100,
                        mean(Difference < 0) * 100)
  ) %>%
  ungroup() %>%
  # Add significance stars based on posterior probability
  mutate(
    Star = ifelse(Post.Prob > 95 | Post.Prob < 5, "*", ""),
    Estimate = round(Estimate, 3),
    Est.Error = round(Est.Error, 3),
    CI.Lower = round(CI.Lower, 3),
    CI.Upper = round(CI.Upper, 3),
    Post.Prob = round(Post.Prob, 2)
  ) %>%
  # Select and arrange the final columns
  select(Comparison, Estimate, Est.Error, CI.Lower, CI.Upper, Post.Prob, Star) %>%
  arrange(Comparison)
  • diff_1_2: Estimate = -0.058 [-0.096, -0.021], Posterior Probability = 99.9%; There is strong evidence that the f0_slopePostCat value increases from percProm1 to percProm2. The negative estimate, combined with a high posterior probability and a credible interval that excludes zero, supports this finding.

  • diff_1_3: Estimate = 0.083 [0.036, 0.128], Posterior Probability = 99.9%; There is strong evidence that f0_slopePostCat is lower at percProm3 than at percProm1. The positive estimate, high posterior probability, and credible interval entirely above zero indicate a reliable increase.

  • diff_2_3: Estimate = 0.141 [0.095, 0.186], Posterior Probability = 100%; There is strong evidence that f0_slopePostCat decreases from percProm2 to percProm3. The positive estimate and a credible interval that does not include zero affirm this upward trend.

Overall Implications:

  • Trend: The findings suggest a non-linear trend in f0_slopePostCat across prominence levels, with a increase from percProm1 to percProm2 followed by a consistent decrease from percProm1 (and percProm2) to percProm3.

  • Interpretation: The posterior probabilities and credible intervals indicate strong support for these changes across the prominence levels, suggesting that f0_slopePostCat varies significantly in response to perceived prominence, with the largest decrease observed between percProm2 and percProm3.

4.1.3.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

data_prepost_cat %>%
  data_grid(percProm = unique(percProm),
            gender_s = 0.1563) %>% # Set to average of gender
  add_epred_draws(mdl_f0_slopePostCat, re_formula = NA) %>%
  mutate(
    percProm_num = as.numeric(as.character(percProm)),
    percProm = factor(percProm, levels = c("1", "2", "3"))
  ) %>%
  ggplot(aes(x = percProm_num, y = .epred, color = percProm)) +
  geom_violin(aes(group = percProm_num, fill = percProm), alpha = 0.8) +
  scale_color_manual(values = colorBlindBlack8) +
  scale_fill_manual(values = colorBlindBlack8) +
  stat_summary(fun = median, geom = "point", color = colorBlindBlack8[7], size = 2) +
  geom_smooth(
    aes(group = 1),
    method = "gam",
    formula = y ~ s(x, bs = "cs", k = 3),
    color = colorBlindBlack8[7],
    se = FALSE
  ) +
  scale_x_continuous(
    breaks = 0:3,
    labels = as.character(0:3)
  ) +
  #ylim(1, 5) +
  labs(
    x = "Perceived prominence level",
    y = "Posterior f0 slope (post-tonic)"
  ) +
  theme_minimal()

post_f0_slopePostCat <- 
  data_prepost_cat %>%
  data_grid(
    percProm = unique(percProm),
    gender_s = 0.1563  # Set gender_s to zero to average over gender
  ) %>%
  add_epred_draws(mdl_f0_slopePostCat, re_formula = NA) %>%
  mutate(
    percProm = factor(percProm, levels = c("1", "2", "3"))
  ) %>%
  ggplot(aes(y = percProm, x = .epred, fill = percProm)) +
  ggdist::stat_halfeye(
    adjust = 0.5,
    width = 0.6,
    .width = c(0.66, 0.95),
    justification = -0.1,
    point_colour = NA
  ) +
  geom_boxplot(
    width = 0.15,
    outlier.shape = NA,
    alpha = 0.5,
    position = position_nudge(y = 0.2)
  ) +
  stat_summary(
    fun = median,
    geom = "point",
    color = "red",
    size = 2,
    position = position_nudge(y = 0.2)
  ) +
  scale_fill_manual(values = colorBlindBlack8) +
  coord_flip() +
  theme_minimal() +
  theme(text = element_text(size = 18)) +
  #xlim(1, 3.5) +
  labs(
    y = "Perceived prominence level",
    x = "Posterior f0 slope (post-tonic)",
    fill = "Perceived\nprominence"
  )

post_f0_slopePostCat;

ggsave(plot = post_f0_slopePostCat, filename = paste0(plots, "posterior_f0_slopePostCat.pdf"), 
       width = 8, height = 6);
ggsave(plot = post_f0_slopePostCat, filename = paste0(plots, "posterior_f0_slopePostCat.jpg"), 
       width = 8, height = 6);
ggsave(plot = post_f0_slopePostCat, filename = paste0(plots, "posterior_f0_slopePostCat.tif"), 
       width = 8, height = 6, compression="lzw", dpi=600);

4.1.3.4 Group-level effects

First, get a general overview.

as.data.frame(VarCorr(mdl_f0_slopePostCat)$participant$sd) %>%
  rownames_to_column(var = "Random Effect") %>%
  select(`Random Effect`, Estimate, Est.Error, Q2.5, Q97.5)
  • percProm1: The estimated standard deviation of random effects at percProm1 is 0.027 (95% CrI: 0.001 to 0.063), suggesting limited variability among participants’ responses at this level. Although small, the credible interval indicates some uncertainty, allowing for variability in individual responses.

  • percProm2: The estimated standard deviation at percProm2 is 0.084 (95% CrI: 0.062 to 0.113), indicating moderate variability among participants. The narrower credible interval compared to percProm1 suggests that the variation in response is more consistently observed at this level.

  • percProm3: The estimated standard deviation at percProm3 is 0.088 (95% CrI: 0.046 to 0.138), indicating moderate variability. The credible interval highlights that participants show more individualized responses at this level, which could be due to different interpretations of prominence.

Overall Implications:

  • Trend: Participant variability in f0_slopePostCat tends to increase as the prominence level rises from percProm1 to percProm3.

  • Interpretation: These results suggest that responses at higher prominence levels (percProm2 and percProm3) exhibit greater participant-specific variability, possibly reflecting more nuanced individual differences in response to perceived prominence at these levels.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

# Extract samples for each participant x percProm
data_prepost_cat %>% 
  data_grid(
    participant = unique(participant),
    percProm = unique(percProm),
    gender_s = 0.1563 # Average over gender
  ) %>% 
  add_epred_draws(mdl_f0_slopePostCat) %>% 
  median_qi()
# Plot the draws for all participants x percProm levels
data_prepost_cat %>%
  # Start by selecting unique participants and their gender_s
  select(participant, gender_s) %>%
  distinct() %>%
  # Back-transform gender_s to gender labels
  mutate(
    gender = if_else(gender_s == -0.5, "Male", "Female")
  ) %>%
  # Combine with percProm levels
  crossing(
    percProm = unique(data_prepost_cat$percProm)
  ) %>%
  # Generate posterior predictions
  add_epred_draws(mdl_f0_slopePostCat) %>%
  # Summarize predictions with .66 and .95 intervals
  median_qi(.width = c(.66, .95)) %>%
  ggplot(aes(x = .epred, y = participant, color = percProm, shape = gender)) +
  # Add the thick line for 66% credible interval
  geom_linerange(aes(xmin = .lower, xmax = .upper, group = .width), 
                 data = ~ filter(., .width == 0.66),  # Use 66% credible interval
                 size = 1,
                 alpha = 0.8,
                 position = position_dodge2(width = 0.7, padding = 0.5)) +
  # Add the thin line for 95% credible interval
  geom_linerange(aes(xmin = .lower, xmax = .upper, group = .width), 
                 data = ~ filter(., .width == 0.95),  # Use 95% credible interval
                 size = 0.5,
                 alpha = 0.8,
                 position = position_dodge2(width = 0.7, padding = 0.5)) +
  # Add points with shapes
  geom_point(position = position_dodge(width = 0.7), size = 2.5) +
  scale_color_manual(values = colorBlindBlack8) +
  labs(
    x = "Posterior f0 slope (post-tonic)",
    y = "Participant",
    color = "Perceived prominence",
    shape = "Gender"
  ) +
  theme_minimal() +
  facet_wrap(~ participant, ncol = 3, scales = "free_y") +
  theme(
    strip.text = element_blank(),
    panel.spacing = unit(0, "lines")
  )

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

# We focus on 95% CrI, but this can be adapted in the width parameter
mdl_f0_slopePostCat %>% 
  spread_draws(r_participant[participant,percProm]) %>% 
  median_qi(r_participant, .width = c(.95))
# In log without Intercept
mdl_f0_slopePostCat %>% 
  spread_draws(r_participant[participant,percProm]) %>% 
  median_qi(r_participant, .width = c(.95, .8, .5)) %>%
  ggplot(aes(y = as.factor(participant), x = r_participant, color = percProm, xmin = .lower, xmax = .upper)) +
  geom_pointinterval(
    position = "dodge") +
  scale_color_manual(
    values = colorBlindBlack8) +
  labs(
    x = "Individual deviations in f0 slope (post-tonic)",
    y = "Participant",
    color = "Perceived prominence") +
  theme_minimal() +
  facet_wrap(~ participant, ncol = 3, scales = "free_y") +
  theme(
    strip.text = element_blank(),
    panel.spacing = unit(0, "lines") 
  )

4.1.4 Frequency modulation median (post-tonic)

4.1.4.1 Priors

4.1.4.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.1697246 0.03077909 0.1188154 0.2373562

An R² of approximately 0.17 suggests a modest fit, indicating that the model explains about 17% of the variance in the dependent variable fmDep_medianPostCat. This means that just under one-fifth of the variability in fmDep_medianPostCat is accounted for by the predictors. The 95% credible interval, which ranges from 11.9% to 23.7%, reflects some uncertainty around the exact R² value but still suggests a modest level of explanatory power for the model.

4.1.4.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: fmDep_medianPost ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_cat (Number of observations: 1568) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1)                1.73      0.22     1.36     2.22 1.00     4168
## sd(percProm2)                1.45      0.16     1.18     1.80 1.00     3678
## sd(percProm3)                1.46      0.21     1.12     1.93 1.00     5291
## cor(percProm1,percProm2)     0.91      0.04     0.81     0.97 1.00     4828
## cor(percProm1,percProm3)     0.96      0.03     0.87     1.00 1.00     5600
## cor(percProm2,percProm3)     0.96      0.03     0.89     0.99 1.00     9874
##                          Tail_ESS
## sd(percProm1)                7223
## sd(percProm2)                6821
## sd(percProm3)                8810
## cor(percProm1,percProm2)     7980
## cor(percProm1,percProm3)     7270
## cor(percProm2,percProm3)    12723
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1     0.17      0.16     0.00     0.58 1.00    11132     9714
## percProm2     0.06      0.06     0.00     0.22 1.00    17647     9086
## percProm3     0.19      0.16     0.01     0.61 1.00    15008     9894
## gender_s      0.13      0.12     0.00     0.45 1.00    12949     9816
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     1.15      0.02     1.10     1.19 1.00    27409    11229
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##           Estimate Est.Error     Q2.5    Q97.5
## percProm1 1.185990  1.171717 1.004751 1.783663
## percProm2 1.060646  1.059923 1.001607 1.242110
## percProm3 1.204865  1.179075 1.005220 1.832147
## gender_s  1.135661  1.129941 1.003480 1.565454

It shows a decrease from 1 to 2 and then increase from 2 to 3. 1 and 3 are the same.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##  0.0007  0.1037  0.2250  0.4453  0.4770  6.1503     678
4.1.4.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  1          1.14         1      1.63
##  2          1.05         1      1.20
##  3          1.16         1      1.67
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm1 - percProm2   0.0794    -0.198     0.631
##  percProm1 - percProm3  -0.0130    -0.683     0.615
##  percProm2 - percProm3  -0.0995    -0.665     0.195
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.1.4.3.2 Hypothesis testing
  • diff_1_2: Estimate = 0.112 [-0.126, 0.532], Posterior Probability = 75.4%; There is moderate evidence suggesting that fmDep_medianPostCat at percProm2 may be lower than at percProm1. However, the credible interval includes zero, indicating that this difference is not reliable.

  • diff_1_3: Estimate = -0.016 [-0.474, 0.455], Posterior Probability = 52.9%; There is no strong evidence for a difference between percProm1 and percProm3. The posterior probability is near chance, and the wide credible interval includes zero, making the difference inconclusive.

  • diff_2_3: Estimate = -0.127 [-0.552, 0.124], Posterior Probability = 78.2%; There is moderate evidence that fmDep_medianPostCat at percProm3 might be higher than at percProm2, but the credible interval includes zero, suggesting that this difference is not robust.

Overall Implications:

  • Trend: There is no clear, strong evidence of reliable differences in fmDep_medianPostCat between the percProm levels. While there is moderate evidence for some comparisons, the inclusion of zero in all credible intervals indicates that the results are inconclusive.
4.1.4.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.1.4.4 Group-level effects

First, get a general overview.

  • percProm1: The estimated standard deviation is 1.732 (95% CrI: 1.359 to 2.221), suggesting substantial variability in participants’ responses at percProm1.

  • percProm2: The estimated standard deviation is 1.451 (95% CrI: 1.179 to 1.798), indicating moderate variability at percProm2, which is slightly lower than at percProm1.

  • percProm3: The estimated standard deviation is 1.457 (95% CrI: 1.116 to 1.929), showing similar variability to percProm2, though with a slightly wider credible interval, suggesting some uncertainty in participants’ responses at this level.

These results highlight that there is considerable participant-level variability in fmDep_medianPostCat across perceived prominence levels, with percProm1 showing the greatest variability.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.1.5 Shannon entropy SD

4.1.5.1 Priors

4.1.5.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2572849 0.01543586 0.2273162 0.2876225

An R² of approximately 0.26 suggests a modest fit. This value indicates that the model explains about 25.7% of the variance in the dependent variable entropySh_sd. In other words, slightly more than a quarter of the variability in entropySh_sd is accounted for by the predictors in the model. The 95% credible interval, ranging from 22.7% to 28.8%, shows some uncertainty around this estimate, but it provides a stable indication of the model’s explanatory power.

4.1.5.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: entropySh_sd ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_cat (Number of observations: 2245) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1)                2.74      0.23     2.31     3.23 1.00      987
## sd(percProm2)                2.61      0.21     2.22     3.05 1.01      922
## sd(percProm3)                2.69      0.24     2.25     3.20 1.00     1180
## cor(percProm1,percProm2)     1.00      0.00     0.99     1.00 1.00     4233
## cor(percProm1,percProm3)     1.00      0.00     0.99     1.00 1.00     4642
## cor(percProm2,percProm3)     1.00      0.00     1.00     1.00 1.00     5539
##                          Tail_ESS
## sd(percProm1)                1896
## sd(percProm2)                1874
## sd(percProm3)                2310
## cor(percProm1,percProm2)     8042
## cor(percProm1,percProm3)     8224
## cor(percProm2,percProm3)    10058
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1     0.16      0.14     0.00     0.54 1.00    15015     8945
## percProm2     0.11      0.11     0.00     0.40 1.00    16910     8873
## percProm3     0.20      0.18     0.01     0.66 1.00    12134     7816
## gender_s      0.28      0.24     0.01     0.91 1.00     3024     3170
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.48      0.01     0.47     0.50 1.00    23597    11899
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##           Estimate Est.Error     Q2.5    Q97.5
## percProm1 1.169202  1.155980 1.004436 1.714580
## percProm2 1.120618  1.113236 1.002985 1.489015
## percProm3 1.222719  1.196302 1.005946 1.940732
## gender_s  1.322735  1.275764 1.008983 2.472945

It shows a decrease from 1 to 2 and then an increase from 2 to 3, and 3 is also sligthly higher than 1.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.     NA's 
## 0.005668 0.026568 0.041266 0.044946 0.058181 0.179098        1
4.1.5.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  1          1.14         1      1.60
##  2          1.11         1      1.42
##  3          1.19         1      1.78
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm1 - percProm2   0.0278    -0.385     0.573
##  percProm1 - percProm3  -0.0318    -0.782     0.592
##  percProm2 - percProm3  -0.0633    -0.736     0.387
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.1.5.3.2 Hypothesis testing
  • diff_1_2: Estimate = 0.042 [-0.281, 0.44], Posterior Probability = 58.6%; There is no strong evidence for a difference between percProm1 and percProm2. The credible interval includes zero, and the posterior probability is close to chance, making the difference unreliable.

  • diff_1_3: Estimate = -0.045 [-0.55, 0.407], Posterior Probability = 57.0%; There is weak evidence suggesting that entropySh_sd at percProm3 is higher than at percProm1. However, the wide credible interval and posterior probability indicate a lack of reliability for this difference.

  • diff_2_3: Estimate = -0.087 [-0.562, 0.253], Posterior Probability = 65.9%; There is weak evidence that entropySh_sd at percProm3 is higher than at percProm2. This difference remains inconclusive, given the wide credible interval and posterior probability below 95%.

Overall Implications:

  • Trend: There is no strong or consistent evidence of significant differences between perceived prominence levels for entropySh_sd. The posterior probabilities and wide credible intervals suggest that any potential differences are uncertain and should be interpreted with caution.
4.1.5.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.1.5.4 Group-level effects

First, get a general overview.

  • percProm1: The estimated standard deviation is 2.736 (95% CrI: 2.312 to 3.229), indicating considerable variability in participants’ responses at percProm1.

  • percProm2: The estimated standard deviation is 2.612 (95% CrI: 2.224 to 3.052), suggesting moderate variability at percProm2, which is slightly lower than at percProm1.

  • percProm3: The estimated standard deviation is 2.687 (95% CrI: 2.254 to 3.198), showing similar variability to percProm1, though with a slightly wider credible interval, suggesting some uncertainty in participants’ responses at this level.

These results highlight that there is considerable participant-level variability in entropySh_sd across perceived prominence levels, with relatively high variability observed at each level.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.1.6 Flux median (post-tonic)

4.1.6.1 Priors

4.1.6.2 Model

Calculate R².

##      Estimate   Est.Error       Q2.5      Q97.5
## R2 0.02975889 0.003809218 0.02285575 0.03776843

An R² of approximately 0.03 suggests a very low fit. This value indicates that the model explains only about 3% of the variance in the dependent variable flux_medianPost. In other words, nearly all of the variability in flux_medianPost is unaccounted for by the predictors in the model. The 95% credible interval, ranging from 2.3% to 3.8%, confirms the minimal explanatory power of the model, indicating high uncertainty in the relationship between the predictors and the response variable.

4.1.6.3 Fixed effects

Let’s check how it looks like.

##  Family: zero_inflated_beta 
##   Links: mu = logit; phi = identity; zi = identity 
## Formula: flux_medianPost ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_cat (Number of observations: 1995) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1)                3.19      0.27     2.70     3.78 1.01     1259
## sd(percProm2)                2.88      0.22     2.48     3.35 1.01      933
## sd(percProm3)                3.02      0.27     2.54     3.61 1.00     1492
## cor(percProm1,percProm2)     1.00      0.00     0.99     1.00 1.00     5065
## cor(percProm1,percProm3)     0.99      0.01     0.98     1.00 1.00     2878
## cor(percProm2,percProm3)     0.99      0.00     0.98     1.00 1.00     6131
##                          Tail_ESS
## sd(percProm1)                2297
## sd(percProm2)                2050
## sd(percProm3)                2780
## cor(percProm1,percProm2)     8158
## cor(percProm1,percProm3)     6586
## cor(percProm2,percProm3)    10528
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1     0.24      0.22     0.01     0.81 1.00    11696     8382
## percProm2     0.10      0.10     0.00     0.37 1.00    15242     8027
## percProm3     0.27      0.25     0.01     0.94 1.00    11329     8841
## gender_s      0.22      0.21     0.01     0.77 1.00     4738     4152
## 
## Further Distributional Parameters:
##     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## phi    18.82      0.80    17.27    20.41 1.00    22417    10691
## zi      0.00      0.00     0.00     0.00 1.00    20690     9102
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##            Estimate Est.Error        Q2.5     Q97.5
## percProm1 0.2356616 0.2208363 0.006266123 0.8114322
## percProm2 0.1000393 0.1004706 0.002549701 0.3706760
## percProm3 0.2709642 0.2494974 0.007335838 0.9388155
## gender_s  0.2172763 0.2066923 0.006234345 0.7657176

It shows a decrease from 1 to 2, and then an increase from 2 to 3. 3 is more or less as high as 1.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
## 0.00000 0.00410 0.00838 0.01712 0.01652 0.99998     251
4.1.6.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  1        0.1712  5.01e-05     0.675
##  2        0.0689  1.89e-05     0.302
##  3        0.2002  2.20e-05     0.773
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## Results are given on the logit (not the response) scale. 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm1 - percProm2   0.0858    -0.287     0.673
##  percProm1 - percProm3  -0.0182    -0.725     0.658
##  percProm2 - percProm3  -0.1110    -0.784     0.270
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## Results are given on the log odds ratio (not the response) scale. 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.1.6.3.2 Hypothesis testing
  • diff_1_2: Estimate = 0.136 [-0.249, 0.734], Posterior Probability = 71.4%; There is weak evidence suggesting that flux_medianPost at percProm2 is lower than at percProm1. However, with a credible interval that includes zero and a posterior probability well below 95%, this difference remains inconclusive.

  • diff_1_3: Estimate = -0.035 [-0.763, 0.627], Posterior Probability = 53.5%; There is no strong evidence for a difference between percProm1 and percProm3, with a posterior probability close to chance.

  • diff_2_3: Estimate = -0.171 [-0.85, 0.232], Posterior Probability = 74.4%; There is weak evidence that flux_medianPost at percProm3 is higher than at percProm2, but this result is inconclusive due to the credible interval including zero.

Overall Implications:

  • Trend: There is no consistent or strong evidence of significant differences between perceived prominence levels for flux_medianPost. The posterior probabilities and wide credible intervals suggest that any potential differences are uncertain and lack reliability.
4.1.6.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.1.6.4 Group-level effects

First, get a general overview.

  • percProm1: The estimated standard deviation is 3.185 (95% CrI: 2.705 to 3.780), indicating considerable variability in participants’ responses at percProm1.

  • percProm2: The estimated standard deviation is 2.875 (95% CrI: 2.477 to 3.352), suggesting moderate participant-level variability at percProm2, slightly lower than at percProm1.

  • percProm3: The estimated standard deviation is 3.016 (95% CrI: 2.541 to 3.611), showing similar variability to percProm1 and percProm2.

These results highlight substantial participant-level variability in flux_medianPost across perceived prominence levels, suggesting that individual differences play a significant role in the responses at each prominence level.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.1.7 Amplitude median (post-tonic)

4.1.7.1 Priors

4.1.7.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5    Q97.5
## R2 0.2911882 0.01956155 0.2538602 0.330177

An R² of approximately 0.29 suggests a moderate fit, indicating that the model explains about 29.1% of the variance in the dependent variable ampl_medianPost. Thus, nearly a third of the variability in ampl_medianPost is accounted for by the predictors. The 95% credible interval (25.4% to 33.0%) indicates moderate certainty around this explanatory power.

4.1.7.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: ampl_medianPost ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_cat (Number of observations: 1995) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1)                2.42      0.23     2.01     2.91 1.00     1550
## sd(percProm2)                2.18      0.21     1.81     2.63 1.00     1599
## sd(percProm3)                2.36      0.23     1.96     2.87 1.00     1789
## cor(percProm1,percProm2)     0.98      0.01     0.97     0.99 1.00     3695
## cor(percProm1,percProm3)     0.97      0.01     0.93     0.99 1.00     3999
## cor(percProm2,percProm3)     0.99      0.00     0.98     1.00 1.00     6398
##                          Tail_ESS
## sd(percProm1)                3003
## sd(percProm2)                3257
## sd(percProm3)                4104
## cor(percProm1,percProm2)     8031
## cor(percProm1,percProm3)     7858
## cor(percProm2,percProm3)    10916
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1     0.12      0.12     0.00     0.44 1.00    16433     9810
## percProm2     0.13      0.12     0.00     0.45 1.00    16053     8821
## percProm3     0.16      0.15     0.00     0.54 1.00    18455    10168
## gender_s      0.23      0.20     0.01     0.76 1.00     5563     5200
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.67      0.01     0.65     0.69 1.00    31800    11661
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##           Estimate Est.Error     Q2.5    Q97.5
## percProm1 1.129096  1.126741 1.003055 1.551368
## percProm2 1.144207  1.127858 1.004314 1.563297
## percProm3 1.173223  1.157395 1.004957 1.718642
## gender_s  1.254358  1.226229 1.006970 2.132875

It shows an increase from 1 to 2, and then a decrease from 2 to 3.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
## 0.00330 0.04785 0.08166 0.10006 0.13731 0.34435     251
4.1.7.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  1          1.10         1      1.46
##  2          1.12         1      1.49
##  3          1.14         1      1.59
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm1 - percProm2  -0.0118    -0.438     0.448
##  percProm1 - percProm3  -0.0278    -0.596     0.440
##  percProm2 - percProm3  -0.0157    -0.559     0.429
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.1.7.3.2 Hypothesis testing
  • diff_1_2: Estimate = -0.013 [-0.357, 0.331], Posterior Probability = 54.2%; There is no strong evidence for a difference between percProm1 and percProm2, with the posterior probability close to chance.

  • diff_1_3: Estimate = -0.038 [-0.445, 0.334], Posterior Probability = 58.0%; There is weak evidence suggesting ampl_medianPost at percProm3 may be slightly lower than percProm1, but it remains inconclusive.

  • diff_2_3: Estimate = -0.025 [-0.424, 0.327], Posterior Probability = 54.7%; No strong evidence for a difference between percProm2 and percProm3, with the posterior probability close to chance.

Overall Implications: There is no strong evidence of consistent differences in ampl_medianPost across perceived prominence levels, indicating that the predictor levels do not significantly impact ampl_medianPost.

4.1.7.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.1.7.4 Group-level effects

First, get a general overview.

  • percProm1: Estimated SD = 2.421 (95% CrI: 2.014 to 2.913)

  • percProm2: Estimated SD = 2.181 (95% CrI: 1.809 to 2.630)

  • percProm3: Estimated SD = 2.363 (95% CrI: 1.963 to 2.875)

Substantial variability is evident across levels, suggesting individual differences in responses to each prominence level.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.1.8 Spectral centroid median

4.1.8.1 Priors

4.1.8.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2441985 0.01455366 0.2157455 0.2728663

An R² of approximately 0.24 suggests a modest fit, with the model explaining around 24.4% of the variance in specCentroid_median. The 95% credible interval (21.6% to 27.3%) suggests some certainty in this explanatory power, though room for improvement remains.

4.1.8.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: specCentroid_median ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_cat (Number of observations: 2245) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1)                0.08      0.01     0.06     0.10 1.00     5503
## sd(percProm2)                0.05      0.01     0.04     0.07 1.00     5568
## sd(percProm3)                0.05      0.01     0.04     0.08 1.00     8711
## cor(percProm1,percProm2)     0.73      0.10     0.49     0.89 1.00     5588
## cor(percProm1,percProm3)     0.57      0.17     0.20     0.85 1.00     7142
## cor(percProm2,percProm3)     0.88      0.09     0.66     0.99 1.00     8314
##                          Tail_ESS
## sd(percProm1)                8775
## sd(percProm2)                8328
## sd(percProm3)               11366
## cor(percProm1,percProm2)     9077
## cor(percProm1,percProm3)    10759
## cor(percProm2,percProm3)    10529
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1     5.76      0.01     5.74     5.79 1.00     3827     5694
## percProm2     5.75      0.01     5.73     5.77 1.00     4177     7493
## percProm3     5.72      0.01     5.70     5.74 1.00     5959     8938
## gender_s      0.01      0.01     0.00     0.03 1.00    10159     9555
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.10      0.00     0.09     0.10 1.00    26006    12346
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##             Estimate Est.Error      Q2.5      Q97.5
## percProm1 318.731797  1.014300 309.90812 327.947156
## percProm2 315.211758  1.009359 309.43561 321.095564
## percProm3 304.976537  1.011357 298.23076 311.820390
## gender_s    1.008489  1.007830   1.00025   1.029005

It shows a decrease from 1 to 3.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   217.1   291.6   314.2   316.7   340.7   421.8       1
4.1.8.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  1           319       310       328
##  2           315       310       321
##  3           305       298       312
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm1 - percProm2     3.53     -3.17      10.1
##  percProm1 - percProm3    13.74      4.87      21.8
##  percProm2 - percProm3    10.24      5.37      15.2
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.1.8.3.2 Hypothesis testing
  • diff_1_2: Estimate = 3.54 [-3.07, 10.3], Posterior Probability = 85.7%; Weak evidence that specCentroid_median at percProm2 is lower than at percProm1, but this difference is not conclusive.

  • diff_1_3: Estimate = 13.8 [5.38, 22.4], Posterior Probability = 99.9%; Strong evidence that specCentroid_median is lower at percProm3 than at percProm1.

  • diff_2_3: Estimate = 10.2 [5.26, 15.1], Posterior Probability = 100%; Strong evidence that specCentroid_median is lower at percProm3 than at percProm2.

Overall Implications: A trend is observed, where specCentroid_median decreases with perceived prominence levels, particularly between percProm1 and percProm3.

4.1.8.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.1.8.4 Group-level effects

First, get a general overview.

  • percProm1: Estimated SD = 0.078 (95% CrI: 0.058 to 0.103)

  • percProm2: Estimated SD = 0.052 (95% CrI: 0.04 to 0.068)

  • percProm3: Estimated SD = 0.054 (95% CrI: 0.038 to 0.075)

Variability remains moderate across levels, suggesting participant responses vary slightly based on perceived prominence levels.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.1.9 Duration (without silences)

4.1.9.1 Priors

4.1.9.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.1999007 0.01466163 0.1713953 0.2290767

An R² of approximately 0.20 indicates a modest fit, with the model explaining about 20.0% of the variance in duration_noSilence. The credible interval (17.1% to 22.9%) indicates moderate certainty around this explanatory power.

4.1.9.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: duration_noSilence ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_cat (Number of observations: 2245) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1)                0.16      0.03     0.11     0.23 1.00     7334
## sd(percProm2)                0.14      0.02     0.11     0.19 1.00     7138
## sd(percProm3)                0.18      0.03     0.12     0.25 1.00     8836
## cor(percProm1,percProm2)     0.66      0.15     0.32     0.89 1.00     4682
## cor(percProm1,percProm3)     0.63      0.19     0.19     0.93 1.00     5587
## cor(percProm2,percProm3)     0.79      0.12     0.48     0.97 1.00     7956
##                          Tail_ESS
## sd(percProm1)               10567
## sd(percProm2)               11384
## sd(percProm3)               11596
## cor(percProm1,percProm2)     7595
## cor(percProm1,percProm3)     7814
## cor(percProm2,percProm3)     9856
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1     5.05      0.03     4.98     5.11 1.00     6212     9625
## percProm2     5.15      0.03     5.10     5.20 1.00     6246     9761
## percProm3     5.29      0.04     5.21     5.37 1.00     7375    11658
## gender_s      0.08      0.05     0.01     0.18 1.00     6689     7514
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.32      0.00     0.31     0.33 1.00    29897    11437
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##             Estimate Est.Error       Q2.5      Q97.5
## percProm1 155.877398  1.034254 145.868565 166.436640
## percProm2 172.523441  1.027185 163.650036 181.817854
## percProm3 198.881462  1.040217 183.619247 214.274873
## gender_s    1.083926  1.046648   1.006463   1.194102

It shows a stable increase from 1 to 3.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##    55.0   135.0   175.0   187.7   225.0   685.0       1
4.1.9.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  1           156       146       166
##  2           173       164       182
##  3           199       184       214
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm1 - percProm2    -16.6     -25.7     -6.98
##  percProm1 - percProm3    -43.2     -57.9    -29.43
##  percProm2 - percProm3    -26.5     -38.4    -14.11
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.1.9.3.2 Hypothesis testing
  • diff_1_2: Estimate = -16.6 [-26.0, -7.29], Posterior Probability = 99.9%; Strong evidence that duration_noSilence increases from percProm1 to percProm2.

  • diff_1_3: Estimate = -43.1 [-57.6, -29.0], Posterior Probability = 100%; Strong evidence that duration_noSilence increases from percProm1 to percProm3.

  • diff_2_3: Estimate = -26.4 [-38.7, -14.4], Posterior Probability = 100%; Strong evidence that duration_noSilence increases from percProm2 to percProm3.

Overall Implications: duration_noSilence tends to increase as perceived prominence levels increase, with consistent evidence supporting this trend.

4.1.9.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.1.9.4 Group-level effects

First, get a general overview.

  • percProm1: Estimated SD = 0.162 (95% CrI: 0.113 to 0.226)

  • percProm2: Estimated SD = 0.143 (95% CrI: 0.109 to 0.188)

  • percProm3: Estimated SD = 0.177 (95% CrI: 0.117 to 0.251)

Variability shows modest participant-level differences across perceived prominence levels.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.1.10 Duration (pretonic)

4.1.10.1 Priors

4.1.10.2 Model

Calculate R².

##     Estimate Est.Error       Q2.5     Q97.5
## R2 0.1239293 0.0153715 0.09510713 0.1551957

An R² of approximately 0.12 indicates a relatively low fit, suggesting the model explains around 12.4% of the variance in durationPre. The 95% credible interval (9.5% to 15.5%) indicates some uncertainty in the explanatory power of the model.

4.1.10.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: durationPre ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_cat (Number of observations: 2063) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1)                0.13      0.03     0.08     0.20 1.00     5841
## sd(percProm2)                0.10      0.02     0.07     0.13 1.00     7318
## sd(percProm3)                0.22      0.04     0.15     0.30 1.00     7764
## cor(percProm1,percProm2)     0.72      0.17     0.30     0.97 1.00     2783
## cor(percProm1,percProm3)     0.50      0.25    -0.04     0.90 1.00     2691
## cor(percProm2,percProm3)     0.83      0.12     0.53     0.98 1.00     7378
##                          Tail_ESS
## sd(percProm1)                9488
## sd(percProm2)               10653
## sd(percProm3)               10926
## cor(percProm1,percProm2)     5667
## cor(percProm1,percProm3)     4202
## cor(percProm2,percProm3)    11626
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1     4.91      0.03     4.84     4.97 1.00     6898     9925
## percProm2     4.98      0.02     4.94     5.02 1.00     6342    10154
## percProm3     5.13      0.05     5.04     5.22 1.00     7834    10167
## gender_s      0.06      0.03     0.01     0.14 1.00     6377     5080
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.38      0.01     0.37     0.39 1.00    23572    11765
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##             Estimate Est.Error       Q2.5      Q97.5
## percProm1 135.038298  1.033495 126.611126 144.033299
## percProm2 146.197507  1.020698 140.468609 152.149767
## percProm3 168.879341  1.047879 153.745826 184.676868
## gender_s    1.066978  1.035397   1.006147   1.147767

It shows a stable increase from 1 to 3.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##    31.0   116.0   146.0   161.7   191.0   573.0     183
4.1.10.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  1           135       126       144
##  2           146       141       152
##  3           169       154       185
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm1 - percProm2    -11.1     -18.9     -2.89
##  percProm1 - percProm3    -34.0     -49.1    -18.65
##  percProm2 - percProm3    -22.8     -35.9     -9.95
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.1.10.3.2 Hypothesis testing
  • diff_1_2: Estimate = -11.1 [-19.2, -3.09], Posterior Probability = 99.6%; Strong evidence that durationPre increases from percProm1 to percProm2.

  • diff_1_3: Estimate = -34.0 [-49.4, -18.9], Posterior Probability = 100%; Strong evidence that durationPre increases from percProm1 to percProm3.

  • diff_2_3: Estimate = -22.8 [-36.0, -10.0], Posterior Probability = 100%; Strong evidence that durationPre increases from percProm2 to percProm3.

Overall Implications: A consistent pattern is observed where durationPre increases as perceived prominence levels increase, with high confidence in these differences.

4.1.10.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.1.10.4 Group-level effects

First, get a general overview.

  • percProm1: Estimated SD = 0.135 (95% CrI: 0.079 to 0.205)

  • percProm2: Estimated SD = 0.098 (95% CrI: 0.070 to 0.134)

  • percProm3: Estimated SD = 0.219 (95% CrI: 0.149 to 0.305)

The variability is somewhat pronounced at percProm3, suggesting that participants vary in responses to perceived prominence.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.1.11 Amplitude SD

4.1.11.1 Priors

4.1.11.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2717001 0.01602145 0.2411063 0.3034417

An R² of approximately 0.27 suggests a moderate fit, indicating the model explains about 27.2% of the variance in ampl_sd. The credible interval (24.1% to 30.3%) suggests moderate certainty around the explanatory power.

4.1.11.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: ampl_sd ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_cat (Number of observations: 2245) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1)                2.92      0.25     2.48     3.46 1.00      970
## sd(percProm2)                2.71      0.23     2.30     3.20 1.00      966
## sd(percProm3)                2.71      0.23     2.29     3.21 1.00     1033
## cor(percProm1,percProm2)     0.99      0.00     0.99     1.00 1.00     3734
## cor(percProm1,percProm3)     0.99      0.01     0.97     1.00 1.00     4060
## cor(percProm2,percProm3)     1.00      0.00     0.99     1.00 1.00     7442
##                          Tail_ESS
## sd(percProm1)                2013
## sd(percProm2)                1928
## sd(percProm3)                2173
## cor(percProm1,percProm2)     8246
## cor(percProm1,percProm3)     7777
## cor(percProm2,percProm3)    12335
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1     0.15      0.14     0.00     0.52 1.00    13096     9095
## percProm2     0.15      0.13     0.00     0.50 1.00    12439     8293
## percProm3     0.15      0.14     0.00     0.52 1.00    14481     8110
## gender_s      0.24      0.22     0.01     0.83 1.00     3635     3687
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.64      0.01     0.62     0.66 1.00    24073    12312
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##           Estimate Est.Error     Q2.5    Q97.5
## percProm1 1.161439  1.153220 1.003742 1.688306
## percProm2 1.162384  1.142325 1.004364 1.641270
## percProm3 1.165793  1.151592 1.004306 1.686098
## gender_s  1.275089  1.251957 1.006832 2.293520

It shows that the DV stays more or less the same.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.     NA's 
## 0.002609 0.019816 0.036262 0.042747 0.059618 0.146670        1
4.1.11.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  1          1.13         1      1.56
##  2          1.14         1      1.54
##  3          1.14         1      1.57
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast               estimate lower.HPD upper.HPD
##  percProm1 - percProm2 -0.004149    -0.516     0.522
##  percProm1 - percProm3 -0.004562    -0.576     0.565
##  percProm2 - percProm3 -0.000474    -0.527     0.510
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.1.11.3.2 Hypothesis testing
  • diff_1_2: Estimate = -0.001 [-0.377, 0.399], Posterior Probability = 51.3%; No strong evidence for a difference between percProm1 and percProm2.

  • diff_1_3: Estimate = -0.004 [-0.415, 0.423], Posterior Probability = 51.3%; No conclusive evidence for a difference between percProm1 and percProm3.

  • diff_2_3: Estimate = -0.003 [-0.39, 0.375], Posterior Probability = 50.2%; No strong evidence for a difference between percProm2 and percProm3.

Overall Implications: There is no strong evidence of significant differences in ampl_sd across perceived prominence levels, suggesting a stable response across conditions.

4.1.11.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.1.11.4 Group-level effects

First, get a general overview.

  • percProm1: Estimated SD = 2.924 (95% CrI: 2.484 to 3.456)

  • percProm2: Estimated SD = 2.711 (95% CrI: 2.296 to 3.203)

  • percProm3: Estimated SD = 2.710 (95% CrI: 2.294 to 3.212)

Participant-level variability remains substantial across conditions, indicating differences in individual responses.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.2 German

Let’s check the correlations between acoustic features.

Let’s set contrasts for comparisons. (Only for intercept models!)

What is the mean for gender.

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -0.50000 -0.50000  0.50000  0.09727  0.50000  0.50000

We go one by one the ten features from top to bottom.

4.2.1 F0 slope

4.2.1.1 Priors

4.2.1.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.1593904 0.01067803 0.1386337 0.1803713

An R² of approximately 0.16 suggests a weak fit. This means the model explains only about 15.9% of the variance in the dependent variable f0_slopeGer, indicating that the majority of the variability in f0_slopeGer is not accounted for by the predictors in the model. The 95% credible interval, ranging from 13.9% to 18.0%, highlights the uncertainty but confirms that the model’s explanatory power is relatively low.

4.2.1.3 Fixed effects

Let’s check how it looks like.

##  Family: student 
##   Links: mu = identity; sigma = identity; nu = identity 
## Formula: f0_slope ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_ger (Number of observations: 2246) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0)                0.11      0.06     0.02     0.27 1.00     2996
## sd(percProm1)                0.05      0.01     0.03     0.07 1.00     9455
## sd(percProm2)                0.09      0.01     0.07     0.11 1.00     8270
## sd(percProm3)                0.11      0.02     0.07     0.15 1.00    10020
## cor(percProm0,percProm1)     0.11      0.35    -0.60     0.75 1.00     2399
## cor(percProm0,percProm2)    -0.02      0.32    -0.68     0.55 1.00     1757
## cor(percProm1,percProm2)     0.59      0.18     0.18     0.87 1.00     5772
## cor(percProm0,percProm3)    -0.34      0.31    -0.86     0.31 1.00     2591
## cor(percProm1,percProm3)     0.17      0.25    -0.32     0.64 1.00     7457
## cor(percProm2,percProm3)     0.58      0.16     0.23     0.84 1.00    11866
##                          Tail_ESS
## sd(percProm0)                5033
## sd(percProm1)               11365
## sd(percProm2)               11868
## sd(percProm3)               12232
## cor(percProm0,percProm1)     3985
## cor(percProm0,percProm2)     2417
## cor(percProm1,percProm2)     9302
## cor(percProm0,percProm3)     5257
## cor(percProm1,percProm3)    10483
## cor(percProm2,percProm3)    13406
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0    -0.06      0.05    -0.15     0.04 1.00     8742     7313
## percProm1    -0.00      0.01    -0.02     0.02 1.00    11442    12028
## percProm2     0.12      0.02     0.09     0.15 1.00     9477    10920
## percProm3     0.19      0.02     0.14     0.23 1.00    12440    11779
## gender_s     -0.03      0.02    -0.07     0.01 1.00    10242    11207
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.14      0.00     0.13     0.15 1.00    20735    12679
## nu        3.05      0.23     2.63     3.54 1.00    22918    12406
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##                Estimate  Est.Error        Q2.5      Q97.5
## percProm0 -0.0563990095 0.04852796 -0.14768697 0.04369150
## percProm1 -0.0001312484 0.01121977 -0.02217870 0.02227687
## percProm2  0.1232162382 0.01573376  0.09214329 0.15437697
## percProm3  0.1874144619 0.02184852  0.14323515 0.23029715
## gender_s  -0.0273822851 0.01996702 -0.06684604 0.01188390

It shows a gradual increase in f0_slope from percProm 0, through 1, 2, up to 3.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.     NA's 
## -2.86180 -0.04065  0.07234  0.09222  0.22678  2.54937       98

Mean is the same, the posterior averages including RE correspond to raw values.

4.2.1.3.1 Effect comparison
##  percProm    emmean lower.HPD upper.HPD
##  0        -0.057886   -0.1518    0.0383
##  1        -0.000132   -0.0224    0.0218
##  2         0.123204    0.0913    0.1535
##  3         0.187518    0.1413    0.2280
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95
##  contrast              estimate lower.HPD upper.HPD
##  percProm0 - percProm1  -0.0574    -0.152    0.0411
##  percProm0 - percProm2  -0.1806    -0.280   -0.0792
##  percProm0 - percProm3  -0.2450    -0.351   -0.1343
##  percProm1 - percProm2  -0.1233    -0.152   -0.0948
##  percProm1 - percProm3  -0.1877    -0.234   -0.1422
##  percProm2 - percProm3  -0.0642    -0.104   -0.0250
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different. For example:

percProm0 - percProm1: The estimated difference in f0_slope between percProm0 and percProm1 is -0.0574. The HPD interval is (-0.152, 0.0411), which includes zero, meaning this difference is not credibly different from zero (i.e., no significant difference between these levels).

percProm0 - percProm2: The estimated difference between percProm0 and percProm2 is -0.1806, and the HPD interval is (-0.280, -0.0792). Since this interval does not include zero, the difference is credibly different from zero, indicating a significant difference in f0_slope between these two levels.

4.2.1.3.2 Hypothesis testing

In our analysis, we are interested in comparing all four levels of perceived prominence (percProm) with each other to determine if there are significant differences in f0_slope across these levels. Since we are using a no-intercept model (i.e., f0_slope ~ 0 + percProm + gender + (0 + percProm | participant)), the fixed effects coefficients directly represent the mean f0_slope for each percProm level.

We choose to use our own custom pipeline instead of the hypothesis() function for the following reasons:

  1. Direct Access to Posterior Samples: By extracting the posterior samples of the fixed effects coefficients, we can directly compute the differences between percProm levels. This approach allows us to utilize the full posterior distribution for each level in our comparisons.

  2. Flexible and Comprehensive Comparisons: Our pipeline enables us to compute all pairwise comparisons between percProm levels in a single, cohesive process. This flexibility is beneficial when dealing with multiple levels and numerous comparisons.

  3. Customized Posterior Probability Calculations: We can calculate the posterior probabilities that the differences are greater than or less than zero, providing a clear measure of the strength of evidence for each comparison.

  4. Simplified Interpretation: The results from our pipeline are presented in an easy-to-interpret format, including estimates, standard errors, credible intervals, and significance indicators, similar to the output of hypothesis() but tailored to our specific model structure.

Using our custom pipeline allows us to:

  • Compute differences between specific levels of percProm directly from the posterior samples of the fixed effects.

  • Obtain posterior probabilities for these differences, giving us insight into the likelihood that a true difference exists.

  • Present the results in a clear and interpretable manner, facilitating straightforward comparison and interpretation.

  • diff_0_1: Estimate = -0.056 [-0.149, 0.045], Posterior Probability = 89.40%; There is moderate evidence that the f0_slope at percProm1 is higher than at percProm0. However, since the credible interval includes zero and the posterior probability is less than 95%, this difference is not considered reliable.

  • diff_0_2: Estimate = -0.180 [-0.278, -0.076], Posterior Probability = 99.60%; Strong evidence that the f0_slope at percProm2 is higher than at percProm0. The negative estimate indicates that percProm0 has a lower f0_slope compared to percProm2.

  • diff_0_3: Estimate = -0.244 [-0.350, -0.134], Posterior Probability = 99.90%; Strong evidence that the f0_slope at percProm3 is higher than at percProm0.

  • diff_1_2: Estimate = -0.123 [-0.153, -0.095], Posterior Probability = 100%; Strong evidence that the f0_slope increases from percProm1 to percProm2.

  • diff_1_3: Estimate = -0.188 [-0.231, -0.141], Posterior Probability = 100%; Strong evidence that the f0_slope increases from percProm1 to percProm3.

  • diff_2_3: Estimate = -0.064 [-0.104, -0.024], Posterior Probability = 99.90%; Strong evidence that the f0_slope increases from percProm2 to percProm3.

Overall Implications:

  • Trend: The f0_slope increases as percProm levels increase from 0 to 3.

  • Interpretation: Except for the comparison between levels 0 and 1, all other comparisons show strong evidence (posterior probability > 95%) of an increase in f0_slope with increasing percProm. As the perceived prominence (percProm) increases, the f0_slope becomes more positive (or less negative), indicating a rising pitch slope. This suggests that higher prominence levels are associated with an increase in f0_slope.

4.2.1.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.2.1.4 Group-level effects

First, get a general overview.

  • percProm0: The estimated standard deviation of the random intercepts is 0.1095 (95% CrI: 0.0168 to 0.2662), indicating moderate variability in participants’ baseline f0_slope.

  • percProm1: The estimated standard deviation is 0.0506 (95% CrI: 0.0306 to 0.0737), suggesting some variability in participants’ response to percProm1.

  • percProm2: The estimated standard deviation is 0.0862 (95% CrI: 0.0656 to 0.1133), indicating some (more that percProm1) variability in participants’ response to percProm2.

  • percProm3: The estimated standard deviation is 0.1052 (95% CrI: 0.0704 to 0.1492), indicating slightly more variability in response to percProm3 compared to percProm2.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.2.2 F0 median

4.2.2.1 Priors

4.2.2.2 Model

Calculate R².

##     Estimate   Est.Error      Q2.5     Q97.5
## R2 0.5735031 0.009071667 0.5551615 0.5906223

An R² of around 0.57 suggests a moderately strong fit. This indicates that the model explains approximately 57.4% of the variance in the dependent variable, meaning more than half of the variability in pitch_medianGer is accounted for by the predictors. The 95% credible interval suggests the R² value is likely between 55.5% and 59.1%, indicating the model’s performance is stable and provides a relatively good explanation of the variance.

4.2.2.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: pitch_median ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_ger (Number of observations: 2156) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0)                0.29      0.08     0.16     0.48 1.00     7315
## sd(percProm1)                0.09      0.01     0.06     0.12 1.00     6263
## sd(percProm2)                0.07      0.01     0.05     0.10 1.00     6426
## sd(percProm3)                0.11      0.02     0.08     0.15 1.00     9066
## cor(percProm0,percProm1)     0.20      0.28    -0.39     0.69 1.00     1904
## cor(percProm0,percProm2)     0.15      0.25    -0.36     0.61 1.00     2334
## cor(percProm1,percProm2)     0.63      0.14     0.32     0.85 1.00     6746
## cor(percProm0,percProm3)    -0.19      0.29    -0.70     0.38 1.00     2741
## cor(percProm1,percProm3)     0.43      0.17     0.06     0.73 1.00     8784
## cor(percProm2,percProm3)     0.72      0.12     0.43     0.91 1.00     9143
##                          Tail_ESS
## sd(percProm0)                9840
## sd(percProm1)                8964
## sd(percProm2)                9368
## sd(percProm3)               12342
## cor(percProm0,percProm1)     3646
## cor(percProm0,percProm2)     4763
## cor(percProm1,percProm2)    10528
## cor(percProm0,percProm3)     6221
## cor(percProm1,percProm3)    11645
## cor(percProm2,percProm3)    13242
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0     5.17      0.09     4.99     5.34 1.00     5311     9008
## percProm1     5.11      0.02     5.08     5.15 1.00     7755     9988
## percProm2     5.17      0.01     5.14     5.20 1.00     8204    10643
## percProm3     5.23      0.02     5.19     5.28 1.00     9812    12084
## gender_s      0.36      0.03     0.31     0.42 1.00     5940     8604
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.15      0.00     0.15     0.16 1.00    27528    11664
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##             Estimate Est.Error       Q2.5      Q97.5
## percProm0 175.184208  1.091839 146.704657 207.733316
## percProm1 166.258631  1.017176 160.676166 171.803876
## percProm2 175.728060  1.013381 171.248364 180.477361
## percProm3 187.510857  1.021714 179.826897 195.759126
## gender_s    1.439071  1.028551   1.360768   1.520318

It shows that pitch median falls from 0 to 1, but raises gradually from 1, through 2 to 3.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   90.55  152.38  191.52  188.92  218.30  475.62     188
4.2.2.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  0           178       148       210
##  1           169       163       175
##  2           179       174       183
##  3           191       183       199
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm0 - percProm1    9.003     -20.0     42.12
##  percProm0 - percProm2   -0.571     -29.9     32.00
##  percProm0 - percProm3  -12.540     -44.1     20.88
##  percProm1 - percProm2   -9.654     -14.6     -4.82
##  percProm1 - percProm3  -21.581     -29.7    -13.79
##  percProm2 - percProm3  -11.960     -18.6     -5.82
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.2.2.3.2 Hypothesis testing
  • diff_0_1: Estimate = 0.052 [-0.124, 0.226], Posterior Probability = 73.90%; There is weak evidence that the pitch_median at percProm1 is higher than at percProm0. However, the posterior probability is low, and the credible interval includes zero, suggesting that this difference is not reliable.

  • diff_0_2: Estimate = -0.003 [-0.179, 0.168], Posterior Probability = 51.40%; No evidence for a difference between percProm0 and percProm2. The estimate is nearly zero, and the credible interval includes zero, making this difference insignificant.

  • diff_0_3: Estimate = -0.068 [-0.251, 0.114], Posterior Probability = 77.90%; There is weak evidence that pitch_median at percProm3 is higher than at percProm0. However, since the credible interval includes zero and the posterior probability is less than 95%, this difference is not considered reliable.

  • diff_1_2: Estimate = -0.055 [-0.084, -0.027], Posterior Probability = 100%; Strong evidence that the pitch_median increases from percProm1 to percProm2. The negative estimate and the credible interval fully below zero support this.

  • diff_1_3: Estimate = -0.120 [-0.164, -0.077], Posterior Probability = 100%; Strong evidence that the pitch_median increases from percProm1 to percProm3.

  • diff_2_3: Estimate = -0.065 [-0.099, -0.032], Posterior Probability = 100%; Strong evidence that the pitch_median increases from percProm2 to percProm3.

Overall Implications:

  • Trend: The pitch_median generally increases as percProm levels increase from 1 to 3. However, there is weak or no evidence of a difference between percProm0 and the other levels.

  • Interpretation: Strong evidence (posterior probability ≥ 95%) is present for increases in pitch_median between levels percProm1, percProm2, and percProm3. However, comparisons involving percProm0 do not show reliable differences.

4.2.2.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.2.2.4 Group-level effects

First, get a general overview.

  • percProm0: The estimated standard deviation of the random effects for percProm0 is 0.2853 (95% CrI: 0.1642 to 0.4785), indicating substantial variability in participants’ baseline pitch_median.

  • percProm1: The estimated standard deviation is 0.0892 (95% CrI: 0.0649 to 0.1211), suggesting relatively low variability in participants’ response to percProm1.

  • percProm2: The estimated standard deviation is 0.0716 (95% CrI: 0.0541 to 0.0951), indicating even lower variability in participants’ response to percProm2.

  • percProm3: The estimated standard deviation is 0.1068 (95% CrI: 0.0771 to 0.1451), indicating moderate variability in participants’ response to percProm3, slightly higher than for percProm2.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.2.3 Amplitude SD

4.2.3.1 Priors

4.2.3.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.5280124 0.01575281 0.4959289 0.5576492

An R² of around 0.53 suggests a moderate fit. The model explains approximately 52.8% of the variance in the dependent variable, meaning just over half of the variability in ampl_sdGer is accounted for by the predictors. The 95% credible interval shows some uncertainty about the exact R² value, but it is likely between 49.6% and 55.8%. This range indicates that the model’s ability to explain the variance is relatively stable.

4.2.3.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: ampl_sd ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_ger (Number of observations: 2344) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0)                3.65      0.27     3.17     4.23 1.00     1802
## sd(percProm1)                3.12      0.21     2.74     3.56 1.00     1625
## sd(percProm2)                2.99      0.21     2.61     3.42 1.00     1795
## sd(percProm3)                2.88      0.23     2.48     3.37 1.00     2245
## cor(percProm0,percProm1)     0.98      0.01     0.96     1.00 1.00     3080
## cor(percProm0,percProm2)     0.98      0.01     0.95     0.99 1.00     2379
## cor(percProm1,percProm2)     1.00      0.00     1.00     1.00 1.00     6207
## cor(percProm0,percProm3)     0.97      0.01     0.94     0.99 1.00     2268
## cor(percProm1,percProm3)     0.99      0.00     0.98     1.00 1.00     5932
## cor(percProm2,percProm3)     1.00      0.00     0.99     1.00 1.00     6950
##                          Tail_ESS
## sd(percProm0)                3236
## sd(percProm1)                2824
## sd(percProm2)                3295
## sd(percProm3)                4186
## cor(percProm0,percProm1)     5671
## cor(percProm0,percProm2)     4535
## cor(percProm1,percProm2)    10932
## cor(percProm0,percProm3)     4524
## cor(percProm1,percProm3)    10486
## cor(percProm2,percProm3)    10815
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0     0.24      0.22     0.01     0.80 1.00    12271     8363
## percProm1     0.11      0.11     0.00     0.40 1.00    16235     8768
## percProm2     0.19      0.15     0.01     0.55 1.00    12832     8215
## percProm3     0.28      0.23     0.01     0.84 1.00    12511     9182
## gender_s      0.24      0.22     0.01     0.81 1.00     7253     6326
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.49      0.01     0.47     0.50 1.00    25941    11257
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##           Estimate Est.Error     Q2.5    Q97.5
## percProm0 1.265566  1.244857 1.005939 2.230485
## percProm1 1.121461  1.114886 1.002993 1.498118
## percProm2 1.203757  1.160954 1.006890 1.737447
## percProm3 1.323594  1.253977 1.009336 2.305958
## gender_s  1.273367  1.240224 1.007432 2.244647

It shows that ampl_sd falls from 0 to 1 and 1 to 2, but raises from 2 to 3 (surpassing 0).

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 0.0006497 0.0075126 0.0123106 0.0156947 0.0202742 0.0821018
4.2.3.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  0          1.20         1      2.00
##  1          1.10         1      1.41
##  2          1.18         1      1.64
##  3          1.27         1      2.09
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm0 - percProm1   0.0839    -0.460     0.971
##  percProm0 - percProm2   0.0245    -0.683     0.917
##  percProm0 - percProm3  -0.0501    -1.130     0.974
##  percProm1 - percProm2  -0.0617    -0.587     0.330
##  percProm1 - percProm3  -0.1515    -1.109     0.354
##  percProm2 - percProm3  -0.0805    -1.008     0.494
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.2.3.3.2 Hypothesis testing
  • diff_0_1: Estimate = 0.121 [-0.277, 0.712], Posterior Probability = 68.10%; There is weak evidence that the ampl_sd at percProm1 is higher than at percProm0. However, the posterior probability is relatively low, and the credible interval includes zero, suggesting that this difference is not reliable.

  • diff_0_2: Estimate = 0.050 [-0.419, 0.666], Posterior Probability = 54.50%; There is no strong evidence for a difference between percProm0 and percProm2. The credible interval spans zero, and the posterior probability is low.

  • diff_0_3: Estimate = -0.045 [-0.686, 0.614], Posterior Probability = 57.00%; There is no clear evidence that the ampl_sd at percProm0 differs from that at percProm3. The posterior probability is too low to support a reliable conclusion.

  • diff_1_2: Estimate = -0.071 [-0.441, 0.252], Posterior Probability = 66.20%; There is moderate but weak evidence suggesting that the ampl_sd at percProm1 is lower than at percProm2, but the credible interval includes zero.

  • diff_1_3: Estimate = -0.166 [-0.738, 0.243], Posterior Probability = 74.90%; There is moderate evidence that the ampl_sd at percProm1 is lower than at percProm3, but this difference is not strong enough to be considered conclusive.

  • diff_2_3: Estimate = -0.095 [-0.639, 0.342], Posterior Probability = 63.70%; There is no clear evidence for a difference between percProm2 and percProm3, as the credible interval includes zero and the posterior probability is moderate.

Overall Implications:

  • Trend: There is no strong evidence of consistent differences in ampl_sd across percProm levels. Most comparisons show weak or inconclusive results, with posterior probabilities below 95%, and credible intervals that include zero.

  • Interpretation: The comparisons do not provide strong support for significant variability in ampl_sd across the perceived prominence levels, suggesting that changes in percProm may not have a substantial impact on ampl_sd.

4.2.3.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.2.3.4 Group-level effects

First, get a general overview.

  • percProm0: The estimated standard deviation of the random effects for percProm0 is 3.650 (95% CrI: 3.167 to 4.229), indicating considerable variability in participants’ ampl_sd at this prominence level.

  • percProm1: The estimated standard deviation for percProm1 is 3.122 (95% CrI: 2.738 to 3.555), suggesting some variability in participants’ ampl_sd at this level, although slightly lower than for percProm0.

  • percProm2: The estimated standard deviation for percProm2 is 2.985 (95% CrI: 2.612 to 3.417), indicating moderate variability in ampl_sd at this prominence level, showing a further decrease compared to percProm1.

  • percProm3: The estimated standard deviation for percProm3 is 2.877 (95% CrI: 2.479 to 3.372), suggesting that the variability in ampl_sd is slightly less than at percProm2, but still present across participants.

Overall Interpretation:

  • Trend: There is a gradual decrease in variability in ampl_sd as the percProm levels increase from 0 to 3. This suggests that participants’ responses in terms of ampl_sd become slightly more consistent as the perceived prominence level increases.

  • Implications: The relatively high standard deviations across all levels indicate that participants’ responses show substantial variability in ampl_sd. However, the decrease in variability from percProm0 to percProm3 suggests that participants may become slightly more uniform in their ampl_sd responses as prominence increases.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.2.4 Amplitude median without silences (post-tonic)

4.2.4.1 Priors

4.2.4.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5    Q97.5
## R2 0.4915381 0.01902245 0.4536859 0.527662

An R² of around 0.49 suggests a moderate fit. This value means that the model explains approximately 49.2% of the variance in the dependent variable. In other words, just under half of the variability in ampl_noSilence_medianPost is accounted for by the predictors in the model. The 95% credible interval indicates that there is some uncertainty about the exact R² value, but it is likely between 45.4% and 52.8%. This range suggests that the model’s ability to explain the variance is relatively stable.

4.2.4.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: ampl_noSilence_medianPost ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_ger (Number of observations: 2277) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0)                3.36      0.27     2.88     3.93 1.00     2351
## sd(percProm1)                2.91      0.21     2.53     3.35 1.00     2192
## sd(percProm2)                2.65      0.20     2.29     3.09 1.00     2368
## sd(percProm3)                2.57      0.23     2.18     3.07 1.00     3285
## cor(percProm0,percProm1)     0.99      0.01     0.96     1.00 1.00     4136
## cor(percProm0,percProm2)     0.98      0.01     0.95     0.99 1.00     3063
## cor(percProm1,percProm2)     0.99      0.00     0.99     1.00 1.00     5511
## cor(percProm0,percProm3)     0.98      0.01     0.96     1.00 1.00     2998
## cor(percProm1,percProm3)     0.99      0.00     0.99     1.00 1.00     6687
## cor(percProm2,percProm3)     1.00      0.00     0.99     1.00 1.00     9274
##                          Tail_ESS
## sd(percProm0)                4593
## sd(percProm1)                4672
## sd(percProm2)                5014
## sd(percProm3)                6675
## cor(percProm0,percProm1)     6025
## cor(percProm0,percProm2)     5786
## cor(percProm1,percProm2)     9518
## cor(percProm0,percProm3)     6222
## cor(percProm1,percProm3)    10780
## cor(percProm2,percProm3)    12666
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0     0.20      0.19     0.01     0.69 1.00    15637     9187
## percProm1     0.12      0.11     0.00     0.41 1.00    17903     8927
## percProm2     0.18      0.16     0.01     0.59 1.00    13404     9825
## percProm3     0.30      0.23     0.01     0.84 1.00    13389     9294
## gender_s      0.24      0.21     0.01     0.80 1.00     8207     7344
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.51      0.01     0.50     0.53 1.00    23873    11410
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##           Estimate Est.Error     Q2.5    Q97.5
## percProm0 1.218554  1.204574 1.005292 1.991287
## percProm1 1.122730  1.118228 1.003054 1.503034
## percProm2 1.199982  1.170568 1.006013 1.803180
## percProm3 1.351038  1.252328 1.012274 2.326486
## gender_s  1.271736  1.239298 1.006549 2.221459

It shows that ampl_noSilence_medianPost falls from 0 to 1 and 1 to 2, but raises from 2 to 3 (surpassing 0).

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
## 0.00203 0.01508 0.02490 0.03119 0.04018 0.18309      67
4.2.4.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  0          1.17         1      1.79
##  1          1.10         1      1.42
##  2          1.17         1      1.67
##  3          1.31         1      2.11
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm0 - percProm1  0.05741    -0.437     0.782
##  percProm0 - percProm2  0.00432    -0.675     0.793
##  percProm0 - percProm3 -0.10798    -1.131     0.748
##  percProm1 - percProm2 -0.05298    -0.653     0.370
##  percProm1 - percProm3 -0.18026    -1.106     0.371
##  percProm2 - percProm3 -0.11998    -0.989     0.523
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.2.4.3.2 Hypothesis testing
  • diff_0_1: Estimate = 0.082 [-0.287, 0.589], Posterior Probability = 64.2%; There is weak evidence that the ampl_noSilence_medianPost at percProm1 is higher than at percProm0. The credible interval includes zero, and the posterior probability is below 95%, so this difference is not considered reliable.

  • diff_0_2: Estimate = 0.015 [-0.459, 0.559], Posterior Probability = 50.8%; There is no strong evidence for a difference between percProm0 and percProm2, with the posterior probability close to chance.

  • diff_0_3: Estimate = -0.103 [-0.706, 0.481], Posterior Probability = 64.8%; There is weak evidence that the ampl_noSilence_medianPost at percProm3 is lower than at percProm0. However, the credible interval includes zero, making the difference unreliable.

  • diff_1_2: Estimate = -0.067 [-0.481, 0.282], Posterior Probability = 64.6%; There is weak evidence that ampl_noSilence_medianPost decreases slightly from percProm1 to percProm2, but the result is not conclusive due to the wide credible interval.

  • diff_1_3: Estimate = -0.185 [-0.747, 0.233], Posterior Probability = 77.8%; There is moderate evidence that ampl_noSilence_medianPost decreases from percProm1 to percProm3. However, since the credible interval includes zero, the evidence remains inconclusive.

  • diff_2_3: Estimate = -0.119 [-0.649, 0.336], Posterior Probability = 68.1%; There is weak evidence that the ampl_noSilence_medianPost decreases slightly from percProm2 to percProm3, but the result is not reliable due to the wide credible interval.

Overall Implications:

  • Trend: There is no strong or consistent evidence of significant differences between percProm levels for the variable ampl_noSilence_medianPost. The posterior probabilities and wide credible intervals suggest that any potential differences are uncertain and should be interpreted with caution.
4.2.4.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.2.4.4 Group-level effects

First, get a general overview.

  • percProm0: The estimated standard deviation of the random intercepts for percProm0 is 3.36 (95% CrI: 2.88 to 3.93), indicating a high level of variability in participants’ baseline ampl_noSilence_medianPost at this level.

  • percProm1: The estimated standard deviation for percProm1 is 2.91 (95% CrI: 2.53 to 3.35), showing moderate variability in participants’ response to percProm1. The range of uncertainty suggests that this variability is stable across participants.

  • percProm2: The estimated standard deviation for percProm2 is 2.65 (95% CrI: 2.29 to 3.09), indicating that participants’ responses to percProm2 are less variable compared to percProm0 and percProm1.

  • percProm3: The estimated standard deviation for percProm3 is 2.57 (95% CrI: 2.18 to 3.07), suggesting a similar degree of variability as percProm2, with slightly lower variability compared to the other percProm levels.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.2.5 Amplitude median without silences

4.2.5.1 Priors

4.2.5.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.5332912 0.01589941 0.5010413 0.5638605

An R² of approximately 0.53 indicates a moderate fit, meaning that the model explains about 53.3% of the variance in the dependent variable, ampl_noSilence_medianGer. In other words, just over half of the variability in ampl_noSilence_medianGer is accounted for by the model’s predictors. The 95% credible interval, ranging from 50.1% to 56.4%, suggests that the model’s explanatory power is relatively stable, with some uncertainty around the exact value. This indicates a solid but not overwhelming predictive performance.

4.2.5.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: ampl_noSilence_median ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_ger (Number of observations: 2344) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0)                3.23      0.26     2.77     3.78 1.00     1610
## sd(percProm1)                2.73      0.20     2.38     3.15 1.00     1413
## sd(percProm2)                2.63      0.20     2.27     3.06 1.00     1542
## sd(percProm3)                2.51      0.22     2.13     2.98 1.00     1979
## cor(percProm0,percProm1)     0.98      0.01     0.96     1.00 1.00     2496
## cor(percProm0,percProm2)     0.99      0.01     0.96     1.00 1.00     2318
## cor(percProm1,percProm2)     1.00      0.00     1.00     1.00 1.00     5830
## cor(percProm0,percProm3)     0.98      0.01     0.96     1.00 1.00     2498
## cor(percProm1,percProm3)     1.00      0.00     0.99     1.00 1.00     6599
## cor(percProm2,percProm3)     1.00      0.00     0.99     1.00 1.00     6514
##                          Tail_ESS
## sd(percProm0)                2511
## sd(percProm1)                2820
## sd(percProm2)                3372
## sd(percProm3)                3560
## cor(percProm0,percProm1)     4187
## cor(percProm0,percProm2)     4089
## cor(percProm1,percProm2)     9640
## cor(percProm0,percProm3)     5156
## cor(percProm1,percProm3)    10787
## cor(percProm2,percProm3)    10749
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0     0.20      0.18     0.01     0.68 1.00    12925     7978
## percProm1     0.10      0.09     0.00     0.33 1.00    15398     8520
## percProm2     0.19      0.15     0.01     0.54 1.00     9577     8445
## percProm3     0.32      0.21     0.01     0.81 1.00     9075     6774
## gender_s      0.23      0.21     0.01     0.77 1.00     6052     6777
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.45      0.01     0.44     0.46 1.00    22738    11936
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##           Estimate Est.Error     Q2.5    Q97.5
## percProm0 1.217194  1.202395 1.005051 1.967473
## percProm1 1.101149  1.094965 1.002825 1.395543
## percProm2 1.205558  1.156491 1.008038 1.716319
## percProm3 1.370609  1.238003 1.014996 2.242720
## gender_s  1.262965  1.227571 1.007569 2.153045

It shows that ampl_noSilence_median falls from 0 to 1 and 1 to 2, but raises from 2 to 3 (surpassing 0).

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## 0.001484 0.017749 0.027888 0.033980 0.042728 0.162021
4.2.5.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  0          1.17         1      1.78
##  1          1.09         1      1.33
##  2          1.18         1      1.62
##  3          1.34         1      2.06
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm0 - percProm1  0.06572    -0.333     0.777
##  percProm0 - percProm2 -0.00519    -0.637     0.701
##  percProm0 - percProm3 -0.14061    -1.112     0.674
##  percProm1 - percProm2 -0.07997    -0.577     0.254
##  percProm1 - percProm3 -0.23328    -1.022     0.250
##  percProm2 - percProm3 -0.13573    -0.926     0.446
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.2.5.3.2 Hypothesis testing
  • diff_0_1: Estimate = 0.10 [-0.222, 0.598], Posterior Probability = 67.4%; There is weak evidence that ampl_noSilence_medianGer at percProm1 is higher than at percProm0. The credible interval includes zero, and the posterior probability is low, suggesting this difference is not reliable.

  • diff_0_2: Estimate = 0.01 [-0.421, 0.534], Posterior Probability = 48.9%; There is no strong evidence for a difference between percProm0 and percProm2, with the estimate very close to zero and the posterior probability reflecting uncertainty.

  • diff_0_3: Estimate = -0.119 [-0.682, 0.462], Posterior Probability = 68.3%; There is weak evidence that ampl_noSilence_medianGer at percProm3 is higher than at percProm0, but the difference is not statistically significant, as the credible interval includes zero.

  • diff_1_2: Estimate = -0.091 [-0.445, 0.192], Posterior Probability = 71.8%; There is weak evidence of a decrease in ampl_noSilence_medianGer from percProm1 to percProm2, but the interval includes zero, and the posterior probability does not reach a conclusive level.

  • diff_1_3: Estimate = -0.219 [-0.714, 0.15], Posterior Probability = 84.8%; There is moderate evidence that ampl_noSilence_medianGer at percProm1 is lower than at percProm3, but the evidence is not strong enough to be reliable.

  • diff_2_3: Estimate = -0.128 [-0.615, 0.292], Posterior Probability = 70.6%; Weak evidence suggests that ampl_noSilence_medianGer increase from percProm2 to percProm3, though this difference is not significant as the interval includes zero.

Overall Interpretation:

  • Trend: There is weak to moderate evidence of differences between prominence levels in ampl_noSilence_medianGer, but none of the comparisons reach a high level of statistical significance (posterior probability > 95%).

  • Interpretation: While some trends suggest a decrease in ampl_noSilence_medianGer as percProm increases, the differences are not strong enough to draw reliable conclusions from these data.

4.2.5.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.2.5.4 Group-level effects

First, get a general overview.

  • percProm0: The estimated standard deviation of the random effects for percProm0 is 3.23 (95% CrI: 2.77 to 3.78), indicating substantial variability between participants at the lowest prominence level.

  • percProm1: The estimated standard deviation for percProm1 is 2.73 (95% CrI: 2.38 to 3.15), suggesting notable variability in participants’ responses to percProm1. The variability is slightly lower than at percProm0.

  • percProm2: The estimated standard deviation for percProm2 is 2.63 (95% CrI: 2.27 to 3.06), indicating some decrease in variability compared to percProm1.

  • percProm3: The estimated standard deviation for percProm3 is 2.51 (95% CrI: 2.13 to 2.98), suggesting that variability in participants’ responses to percProm3 is the lowest among all prominence levels, but still substantial.

Implication: The variability in responses across participants decreases as the perceived prominence level (percProm) increases from percProm0 to percProm3. However, there is still considerable individual variability at all levels, particularly at the lower levels of prominence.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.2.6 Duration

4.2.6.1 Priors

4.2.6.2 Model

Calculate R².

##     Estimate  Est.Error     Q2.5     Q97.5
## R2 0.2538415 0.01672139 0.221985 0.2872682

An R² of approximately 0.25 suggests a modest fit. This value indicates that the model explains about 25.4% of the variance in the dependent variable duration. In other words, a quarter of the variability in duration is accounted for by the predictors in the model. The 95% credible interval, ranging from 22.2% to 28.7%, shows that there is some uncertainty around this estimate, but it provides a stable indication of the model’s explanatory power.

4.2.6.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: duration ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_ger (Number of observations: 2344) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0)                0.12      0.08     0.01     0.29 1.00     3362
## sd(percProm1)                0.12      0.02     0.08     0.16 1.00     8384
## sd(percProm2)                0.16      0.02     0.12     0.21 1.00     6206
## sd(percProm3)                0.24      0.04     0.18     0.31 1.00     7777
## cor(percProm0,percProm1)     0.16      0.35    -0.56     0.79 1.00     1222
## cor(percProm0,percProm2)     0.20      0.35    -0.54     0.79 1.00     1076
## cor(percProm1,percProm2)     0.86      0.08     0.66     0.98 1.00     6454
## cor(percProm0,percProm3)     0.14      0.37    -0.61     0.78 1.00     1263
## cor(percProm1,percProm3)     0.74      0.13     0.44     0.94 1.00     6158
## cor(percProm2,percProm3)     0.91      0.06     0.76     0.99 1.00    10445
##                          Tail_ESS
## sd(percProm0)                5055
## sd(percProm1)               11514
## sd(percProm2)                8900
## sd(percProm3)               10667
## cor(percProm0,percProm1)     2228
## cor(percProm0,percProm2)     1898
## cor(percProm1,percProm2)     9403
## cor(percProm0,percProm3)     2506
## cor(percProm1,percProm3)     9321
## cor(percProm2,percProm3)    13585
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0     5.34      0.05     5.24     5.44 1.00    14198    11429
## percProm1     5.45      0.02     5.40     5.49 1.00     6290     9564
## percProm2     5.56      0.03     5.50     5.62 1.00     5490     8260
## percProm3     5.65      0.05     5.56     5.74 1.00     6540     9298
## gender_s      0.07      0.04     0.01     0.15 1.00     7867     6854
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.30      0.00     0.29     0.31 1.00    27809    11172
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##             Estimate Est.Error       Q2.5      Q97.5
## percProm0 208.212541  1.052832 188.338451 230.704255
## percProm1 231.986769  1.023748 221.536232 243.057688
## percProm2 259.791602  1.029371 245.052276 274.977087
## percProm3 284.051734  1.046606 259.105665 310.231991
## gender_s    1.074742  1.038197   1.007181   1.163993

It shows a stable increase from 0 to 3.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    69.0   205.0   256.0   271.0   315.2   895.0
4.2.6.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  0           208       188       230
##  1           232       221       243
##  2           260       245       275
##  3           285       260       311
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm0 - percProm1    -23.8     -47.4     -0.99
##  percProm0 - percProm2    -51.7     -76.4    -26.98
##  percProm0 - percProm3    -75.9    -109.2    -44.62
##  percProm1 - percProm2    -27.8     -38.2    -17.48
##  percProm1 - percProm3    -52.1     -74.2    -31.66
##  percProm2 - percProm3    -24.4     -41.5     -8.27
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.2.6.3.2 Hypothesis testing
  • diff_0_1: Estimate = -0.108 [-0.217, 0.003], Posterior Probability = 97.3%; There is strong evidence that the duration at percProm1 is higher than at percProm0, with the credible interval slightly crossing zero.

  • diff_0_2: Estimate = -0.221 [-0.332, -0.108], Posterior Probability = 100%; Strong evidence that duration at percProm2 is higher than at percProm0, as indicated by the negative estimate and a credible interval entirely below zero.

  • diff_0_3: Estimate = -0.311 [-0.441, -0.177], Posterior Probability = 100%; Strong evidence that duration at percProm3 is higher than at percProm0.

  • diff_1_2: Estimate = -0.113 [-0.154, -0.073], Posterior Probability = 100%; Strong evidence of an increase in duration from percProm1 to percProm2.

  • diff_1_3: Estimate = -0.202 [-0.276, -0.128], Posterior Probability = 100%; Strong evidence of an increase in duration from percProm1 to percProm3.

  • diff_2_3: Estimate = -0.089 [-0.146, -0.032], Posterior Probability = 99.8%; Strong evidence of an increase in duration from percProm2 to percProm3.

Overall Implications:

  • Trend: The duration increases as percProm levels increase from 0 to 3.

  • Interpretation: All comparisons, except for the borderline case of diff_0_1, show strong evidence (posterior probability > 95%) of an increase in duration with increasing percProm. This suggests that higher prominence levels are associated with a longer duration.

4.2.6.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.2.6.4 Group-level effects

First, get a general overview.

  • percProm0: The estimated standard deviation of the random intercepts for percProm0 is 0.119 (95% CrI: 0.006 to 0.288), suggesting modest variability in participants’ baseline duration at the lowest prominence level.

  • percProm1: The estimated standard deviation is 0.116 (95% CrI: 0.083 to 0.157), indicating some variability in participants’ responses to percProm1.

  • percProm2: The estimated standard deviation is 0.161 (95% CrI: 0.125 to 0.208), showing a slightly higher variability in response to percProm2 compared to percProm1.

  • percProm3: The estimated standard deviation is 0.237 (95% CrI: 0.177 to 0.315), indicating the most variability among participants at the highest prominence level (percProm3).

Implications:

  • The variability in participants’ duration responses increases with higher prominence levels, with the most substantial variation seen at percProm3. This pattern suggests that participants’ realization of higher prominence levels is less uniform.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.2.7 Flux SD

4.2.7.1 Priors

4.2.7.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.4174519 0.03402887 0.3503354 0.4827032

An R² of approximately 0.42 suggests a moderate fit. This value means that the model explains about 41.7% of the variance in the flux_sd variable. In other words, a little over 40% of the variability in flux_sd is accounted for by the predictors in the model. The 95% credible interval indicates some uncertainty about the exact R² value, but it is likely between 35.0% and 48.3%. This range shows that while the model explains a significant portion of the variance, there is still room for improvement in capturing all the variability in flux_sd.

4.2.7.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: flux_sd ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_ger (Number of observations: 2344) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0)                1.98      0.28     1.51     2.60 1.00     5075
## sd(percProm1)                2.22      0.22     1.84     2.72 1.00     5392
## sd(percProm2)                2.29      0.20     1.93     2.73 1.00     5069
## sd(percProm3)                2.47      0.22     2.07     2.92 1.00     5492
## cor(percProm0,percProm1)     0.80      0.09     0.59     0.92 1.00     2659
## cor(percProm0,percProm2)     0.78      0.09     0.56     0.92 1.00     2414
## cor(percProm1,percProm2)     0.99      0.00     0.98     1.00 1.00     7160
## cor(percProm0,percProm3)     0.78      0.09     0.55     0.92 1.00     2463
## cor(percProm1,percProm3)     0.99      0.01     0.97     1.00 1.00     7346
## cor(percProm2,percProm3)     1.00      0.00     0.99     1.00 1.00     9928
##                          Tail_ESS
## sd(percProm0)                8640
## sd(percProm1)                8770
## sd(percProm2)                8736
## sd(percProm3)                9116
## cor(percProm0,percProm1)     5444
## cor(percProm0,percProm2)     5150
## cor(percProm1,percProm2)    10693
## cor(percProm0,percProm3)     4488
## cor(percProm1,percProm3)    11060
## cor(percProm2,percProm3)    13294
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0     0.25      0.23     0.01     0.87 1.00     7348     9628
## percProm1     0.23      0.18     0.01     0.68 1.00    10491     9880
## percProm2     0.14      0.13     0.00     0.48 1.00    15737     8107
## percProm3     0.11      0.11     0.00     0.40 1.00    18965     9098
## gender_s      0.19      0.18     0.01     0.67 1.00    14601     9404
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.92      0.01     0.90     0.95 1.00    27544    11856
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##           Estimate Est.Error     Q2.5    Q97.5
## percProm0 1.286330  1.263735 1.007152 2.382555
## percProm1 1.253080  1.202646 1.008544 1.980475
## percProm2 1.151640  1.136491 1.003820 1.608740
## percProm3 1.115284  1.113929 1.003001 1.488405
## gender_s  1.214464  1.199854 1.005241 1.957386

It shows that flux_sd systematically falls from 0 to 3.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 0.0000145 0.0349014 0.0728800 0.0972661 0.1298478 0.4803707
4.2.7.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  0          1.21         1      2.09
##  1          1.21         1      1.83
##  2          1.12         1      1.51
##  3          1.09         1      1.39
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm0 - percProm1   0.0094    -0.829     1.078
##  percProm0 - percProm2   0.0781    -0.579     1.055
##  percProm0 - percProm3   0.1044    -0.397     1.166
##  percProm1 - percProm2   0.0687    -0.439     0.764
##  percProm1 - percProm3   0.0991    -0.358     0.834
##  percProm2 - percProm3   0.0229    -0.389     0.488
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.2.7.3.2 Hypothesis testing
  • diff_0_1: Estimate = 0.026 [-0.528, 0.697], Posterior Probability = 51.70%; There is no strong evidence for a difference between flux_sd at percProm0 and percProm1. The estimate is positive, but the credible interval includes zero, suggesting uncertainty in this comparison.

  • diff_0_2: Estimate = 0.111 [-0.342, 0.755], Posterior Probability = 64.50%; There is moderate evidence that flux_sd at percProm2 is higher than at percProm0, but the evidence is not strong, as the credible interval still includes zero.

  • diff_0_3: Estimate = 0.143 [-0.262, 0.786], Posterior Probability = 70.60%; There is moderate evidence that flux_sd increases from percProm0 to percProm3, but again, the credible interval includes zero, making the difference not conclusive.

  • diff_1_2: Estimate = 0.084 [-0.303, 0.554], Posterior Probability = 65.50%; Moderate evidence suggests that flux_sd increases slightly from percProm1 to percProm2, but the credible interval includes zero, indicating uncertainty in the direction of the effect.

  • diff_1_3: Estimate = 0.116 [-0.256, 0.597], Posterior Probability = 70.90%; There is moderate evidence that flux_sd increases from percProm1 to percProm3, but the evidence is not strong enough to confirm the increase.

  • diff_2_3: Estimate = 0.032 [-0.293, 0.388], Posterior Probability = 57.70%; There is weak evidence for a difference between percProm2 and percProm3. The credible interval includes zero, indicating considerable uncertainty.

Overall Implications:

  • Trend: While the estimates suggest a possible increase in flux_sd as percProm levels increase, the credible intervals for most comparisons include zero, indicating that the evidence is not strong enough to confirm these differences.

  • Interpretation: The comparisons do not show strong or consistent evidence of an increase in flux_sd across prominence levels. The posterior probabilities are all below 95%, indicating that none of the comparisons can be considered reliable.

4.2.7.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.2.7.4 Group-level effects

First, get a general overview.

  • percProm0: The estimated standard deviation of the random intercepts at percProm0 is 1.984 (95% CrI: 1.508 to 2.595), indicating notable variability in participants’ flux_sd at this level.

  • percProm1: The estimated standard deviation at percProm1 is 2.223 (95% CrI: 1.840 to 2.716), suggesting an increase in variability in participants’ flux_sd compared to percProm0.

  • percProm2: The estimated standard deviation at percProm2 is 2.294 (95% CrI: 1.925 to 2.731), indicating a slight increase in variability compared to percProm1.

  • percProm3: The estimated standard deviation at percProm3 is 2.466 (95% CrI: 2.073 to 2.924), showing further increased variability in participants’ flux_sd compared to the other prominence levels.

Interpretation: The increasing standard deviations across percProm levels suggest that variability in participants’ flux_sd tends to grow as perceived prominence increases. This implies greater individual differences in how participants’ flux_sd changes across different levels of prominence.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.2.8 Flux median

4.2.8.1 Priors

4.2.8.2 Model

Calculate R².

##     Estimate   Est.Error     Q2.5     Q97.5
## R2 0.4894135 0.008934676 0.470584 0.5019241

An R² of 0.489 (95% CrI: 0.470 to 0.501) indicates a moderate fit, meaning that approximately 48.9% of the variance in flux_median is explained by the model. This credible interval suggests a stable estimate within a range of explained variability.

4.2.8.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: flux_median ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_ger (Number of observations: 2344) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0)                5.42      0.41     4.67     6.26 1.00     6848
## sd(percProm1)                3.68      0.26     3.20     4.25 1.00     6845
## sd(percProm2)                3.25      0.22     2.85     3.72 1.00     6385
## sd(percProm3)                3.13      0.24     2.70     3.65 1.00     7524
## cor(percProm0,percProm1)     0.79      0.07     0.63     0.90 1.00     4236
## cor(percProm0,percProm2)     0.79      0.07     0.64     0.90 1.00     3921
## cor(percProm1,percProm2)     0.97      0.01     0.94     0.99 1.00     7237
## cor(percProm0,percProm3)     0.81      0.07     0.64     0.92 1.00     4447
## cor(percProm1,percProm3)     0.94      0.03     0.88     0.98 1.00     8466
## cor(percProm2,percProm3)     0.98      0.01     0.95     1.00 1.00    12206
##                          Tail_ESS
## sd(percProm0)                9794
## sd(percProm1)                8985
## sd(percProm2)                8707
## sd(percProm3)               10362
## cor(percProm0,percProm1)     7222
## cor(percProm0,percProm2)     6422
## cor(percProm1,percProm2)    10596
## cor(percProm0,percProm3)     8103
## cor(percProm1,percProm3)    10931
## cor(percProm2,percProm3)    14385
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0     0.28      0.26     0.01     0.95 1.00    14295     9081
## percProm1     0.25      0.23     0.01     0.87 1.00    16139     9420
## percProm2     0.16      0.16     0.00     0.57 1.00    19134     8938
## percProm3     0.20      0.19     0.01     0.69 1.00    16999     9742
## gender_s      0.28      0.26     0.01     0.98 1.00    18520     9927
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     2.37      0.04     2.30     2.44 1.00    26057    11789
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##           Estimate Est.Error     Q2.5    Q97.5
## percProm0 1.323592  1.295039 1.007500 2.574693
## percProm1 1.280032  1.262147 1.006795 2.384896
## percProm2 1.175126  1.169967 1.004069 1.772865
## percProm3 1.221109  1.207878 1.005733 1.995472
## gender_s  1.325376  1.300302 1.007319 2.677204

It shows that flux_median falls from 0 to 1 and 1 to 2, but raises from 2 to 3.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 8.000e-08 4.676e-03 8.396e-03 1.397e-02 1.600e-02 2.274e-01
4.2.8.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  0          1.25         1      2.26
##  1          1.22         1      2.07
##  2          1.14         1      1.64
##  3          1.18         1      1.82
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm0 - percProm1   0.0242    -1.141     1.346
##  percProm0 - percProm2   0.0868    -0.652     1.311
##  percProm0 - percProm3   0.0544    -0.815     1.362
##  percProm1 - percProm2   0.0584    -0.657     1.111
##  percProm1 - percProm3   0.0320    -0.856     1.138
##  percProm2 - percProm3  -0.0236    -0.844     0.627
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.2.8.3.2 Hypothesis testing
  • diff_0_1: Estimate = 0.033 [-0.670, 0.777], Posterior Probability = 53.7%; There is no strong evidence for a difference between percProm0 and percProm1, with the posterior probability close to chance.

  • diff_0_2: Estimate = 0.119 [-0.415, 0.829], Posterior Probability = 64.0%; There is weak evidence suggesting that flux_median at percProm2 is higher than at percProm0, though the credible interval includes zero, making the difference unreliable.

  • diff_0_3: Estimate = 0.081 [-0.521, 0.807], Posterior Probability = 58.6%; No strong evidence for a difference between percProm0 and percProm3, with a credible interval spanning zero.

  • diff_1_2: Estimate = 0.086 [-0.418, 0.741], Posterior Probability = 60.8%; Weak evidence indicates that flux_median may increase from percProm1 to percProm2, though the result is inconclusive.

  • diff_1_3: Estimate = 0.047 [-0.539, 0.723], Posterior Probability = 55.5%; There is no strong evidence for a difference between percProm1 and percProm3, given the credible interval includes zero.

  • diff_2_3: Estimate = -0.038 [-0.569, 0.441], Posterior Probability = 55.7%; There is weak evidence of a slight decrease in flux_median from percProm2 to percProm3, but the difference is unreliable.

Overall Implications:

  • Trend: There is no strong or consistent evidence of significant differences between percProm levels for flux_median. The posterior probabilities and wide credible intervals suggest that any potential differences are uncertain and should be interpreted cautiously.
4.2.8.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.2.8.4 Group-level effects

First, get a general overview.

  • percProm0: SD = 1.984 (95% CrI: 1.508 to 2.595), indicating moderate variability in participants’ flux_median.

  • percProm1: SD = 2.223 (95% CrI: 1.840 to 2.716), suggesting an increase in variability compared to percProm0.

  • percProm2: SD = 2.294 (95% CrI: 1.925 to 2.731), indicating further increased variability.

  • percProm3: SD = 2.466 (95% CrI: 2.073 to 2.924), showing the highest variability, indicating substantial differences across participants in flux_median.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.2.9 Amplitude SD without silences

4.2.9.1 Priors

4.2.9.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.5297911 0.01536498 0.4984655 0.5587921

An R² of 0.530 (95% CrI: 0.498 to 0.559) indicates a moderate fit, with approximately 53% of the variance in ampl_noSilence_sd explained by the model. The narrow credible interval suggests stability in the estimate.

4.2.9.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: ampl_noSilence_sd ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_ger (Number of observations: 2344) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0)                3.67      0.27     3.18     4.26 1.00     2276
## sd(percProm1)                3.14      0.21     2.75     3.59 1.00     1975
## sd(percProm2)                3.01      0.21     2.62     3.45 1.00     2145
## sd(percProm3)                2.91      0.23     2.50     3.41 1.00     2773
## cor(percProm0,percProm1)     0.98      0.01     0.96     0.99 1.00     3876
## cor(percProm0,percProm2)     0.98      0.01     0.95     0.99 1.00     3335
## cor(percProm1,percProm2)     1.00      0.00     1.00     1.00 1.00     6795
## cor(percProm0,percProm3)     0.97      0.01     0.94     0.99 1.00     3177
## cor(percProm1,percProm3)     0.99      0.00     0.99     1.00 1.00     6122
## cor(percProm2,percProm3)     1.00      0.00     0.99     1.00 1.00     7055
##                          Tail_ESS
## sd(percProm0)                4497
## sd(percProm1)                4288
## sd(percProm2)                4298
## sd(percProm3)                5632
## cor(percProm0,percProm1)     6122
## cor(percProm0,percProm2)     5482
## cor(percProm1,percProm2)    11049
## cor(percProm0,percProm3)     5580
## cor(percProm1,percProm3)     9943
## cor(percProm2,percProm3)    12196
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0     0.23      0.21     0.01     0.78 1.00    16189     9946
## percProm1     0.12      0.11     0.00     0.39 1.00    19139     9704
## percProm2     0.18      0.14     0.01     0.53 1.00    15145     8849
## percProm3     0.27      0.22     0.01     0.80 1.00    17008     9829
## gender_s      0.24      0.22     0.01     0.81 1.00    10463     7318
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.48      0.01     0.46     0.49 1.00    29045    11764
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##           Estimate Est.Error     Q2.5    Q97.5
## percProm0 1.263459  1.237401 1.006980 2.180809
## percProm1 1.122085  1.112960 1.003245 1.482347
## percProm2 1.200015  1.153557 1.007086 1.701875
## percProm3 1.309910  1.240539 1.010086 2.231745
## gender_s  1.276043  1.246767 1.006776 2.238174

It shows that ampl_noSilence_sd falls from 0 to 1 and 1 to 2, but raises from 2 to 3 (surpassing 0).

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## 0.0006497 0.0073900 0.0120082 0.0150229 0.0192005 0.0795982
4.2.9.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  0          1.21         1      1.97
##  1          1.10         1      1.42
##  2          1.18         1      1.61
##  3          1.26         1      2.05
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm0 - percProm1   0.0855    -0.426     0.947
##  percProm0 - percProm2   0.0257    -0.618     0.907
##  percProm0 - percProm3  -0.0385    -1.064     0.926
##  percProm1 - percProm2  -0.0605    -0.566     0.344
##  percProm1 - percProm3  -0.1406    -1.052     0.358
##  percProm2 - percProm3  -0.0754    -0.850     0.542
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.2.9.3.2 Hypothesis testing
  • diff_0_1: Estimate = 0.119 [-0.271, 0.685], Posterior Probability = 68.2%; There is weak evidence that ampl_noSilence_sd at percProm1 is higher than at percProm0, but the result is not conclusive.

  • diff_0_2: Estimate = 0.052 [-0.401, 0.649], Posterior Probability = 55.0%; There is no strong evidence for a difference between percProm0 and percProm2, with the posterior probability close to chance.

  • diff_0_3: Estimate = -0.036 [-0.650, 0.595], Posterior Probability = 55.6%; There is no conclusive evidence of a difference between percProm0 and percProm3, given the wide credible interval.

  • diff_1_2: Estimate = -0.067 [-0.427, 0.261], Posterior Probability = 66.6%; Weak evidence suggests a slight decrease in ampl_noSilence_sd from percProm1 to percProm2, though this difference remains unreliable.

  • diff_1_3: Estimate = -0.155 [-0.707, 0.251], Posterior Probability = 74.5%; Moderate evidence of a decrease in ampl_noSilence_sd from percProm1 to percProm3, but it is not conclusive due to the inclusion of zero.

  • diff_2_3: Estimate = -0.088 [-0.606, 0.329], Posterior Probability = 63.8%; Weak evidence that ampl_noSilence_sd decreases from percProm2 to percProm3, though the result is not reliable.

Overall Implications:

  • Trend: There is no consistent or strong evidence of significant differences between percProm levels for ampl_noSilence_sd. The posterior probabilities and wide credible intervals imply that any potential differences are uncertain and should be interpreted with caution.
4.2.9.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.2.9.4 Group-level effects

First, get a general overview.

  • percProm0: SD = 3.672 (95% CrI: 3.183 to 4.263), indicating high variability in participants’ ampl_noSilence_sd.

  • percProm1: SD = 3.142 (95% CrI: 2.752 to 3.587), suggesting reduced variability.

  • percProm2: SD = 3.009 (95% CrI: 2.625 to 3.451), indicating a slight decrease in variability compared to percProm1.

  • percProm3: SD = 2.905 (95% CrI: 2.504 to 3.408), showing the lowest variability, suggesting participants’ responses are more consistent at this level.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

4.2.10 F0 median (post-tonic)

4.2.10.1 Priors

4.2.10.2 Model

Calculate R².

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.4325174 0.01226329 0.4081115 0.4560081

An R² of 0.433 (95% CrI: 0.408 to 0.456) suggests a moderate fit, with approximately 43.3% of the variance in pitch_medianPost explained by the model. The interval implies a stable estimate within the range.

4.2.10.3 Fixed effects

Let’s check how it looks like.

##  Family: lognormal 
##   Links: mu = identity; sigma = identity 
## Formula: pitch_medianPost ~ 0 + percProm + gender_s + (0 + percProm | participant) 
##    Data: data_prepost_ger (Number of observations: 1821) 
##   Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
##          total post-warmup draws = 16000
## 
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35) 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0)                0.13      0.06     0.02     0.27 1.00     5473
## sd(percProm1)                0.09      0.02     0.07     0.13 1.00     9059
## sd(percProm2)                0.08      0.01     0.06     0.11 1.00     7520
## sd(percProm3)                0.08      0.02     0.05     0.12 1.00     9609
## cor(percProm0,percProm1)     0.30      0.33    -0.42     0.85 1.00     1605
## cor(percProm0,percProm2)     0.14      0.37    -0.58     0.79 1.00     1322
## cor(percProm1,percProm2)     0.68      0.14     0.36     0.90 1.00     6866
## cor(percProm0,percProm3)    -0.02      0.40    -0.74     0.73 1.00     1965
## cor(percProm1,percProm3)     0.55      0.20     0.12     0.87 1.00     9960
## cor(percProm2,percProm3)     0.70      0.16     0.33     0.94 1.00    10002
##                          Tail_ESS
## sd(percProm0)                4012
## sd(percProm1)               11703
## sd(percProm2)               11132
## sd(percProm3)                9856
## cor(percProm0,percProm1)     2687
## cor(percProm0,percProm2)     3253
## cor(percProm1,percProm2)    10418
## cor(percProm0,percProm3)     4592
## cor(percProm1,percProm3)    11718
## cor(percProm2,percProm3)    12171
## 
## Regression Coefficients:
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0     5.13      0.05     5.03     5.24 1.00     7530    10205
## percProm1     5.13      0.02     5.10     5.17 1.00     8390     9919
## percProm2     5.21      0.01     5.18     5.24 1.00     9141    10531
## percProm3     5.21      0.02     5.17     5.25 1.00    10635    12435
## gender_s      0.33      0.03     0.28     0.39 1.00     9148    10611
## 
## Further Distributional Parameters:
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.18      0.00     0.17     0.18 1.00    21258    11305
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
##             Estimate Est.Error       Q2.5      Q97.5
## percProm0 169.820804  1.054882 152.182838 188.107643
## percProm1 169.526200  1.018666 163.403936 175.762420
## percProm2 183.921112  1.015061 178.553955 189.478645
## percProm3 183.263705  1.019559 176.436022 190.420013
## gender_s    1.393311  1.027689   1.321521   1.469917

It shows that pitch_medianPost is the same on 0 and 1, and raises to stay the same on 2 and 3.

Diagnostic plots.

Posterior predictive check.

Extract samples.

Check conditional effect.

Compare conditional effects to raw values.

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   90.61  163.13  192.13  196.89  227.88  450.43     523
4.2.10.3.1 Effect comparison
##  percProm emmean lower.HPD upper.HPD
##  0           172       154       190
##  1           172       166       178
##  2           186       181       192
##  3           186       179       193
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

##  contrast              estimate lower.HPD upper.HPD
##  percProm0 - percProm1    0.370    -19.08     18.20
##  percProm0 - percProm2  -14.105    -32.15      4.95
##  percProm0 - percProm3  -13.421    -33.23      5.73
##  percProm1 - percProm2  -14.610    -20.25     -9.28
##  percProm1 - percProm3  -13.932    -21.39     -6.40
##  percProm2 - percProm3    0.629     -5.63      7.19
## 
## Results are averaged over the levels of: gender_s 
## Point estimate displayed: median 
## HPD interval probability: 0.95

If the comparison includes 0, the contrast is not reliably different.

4.2.10.3.2 Hypothesis testing
  • diff_0_1: Estimate = 0.002 [-0.109, 0.109], Posterior Probability = 51.7%; No strong evidence for a difference between percProm0 and percProm1, with a posterior probability close to chance.

  • diff_0_2: Estimate = -0.08 [-0.190, 0.024], Posterior Probability = 93.5%; Moderate evidence that pitch_medianPost at percProm0 is slightly lower than at percProm2, but not conclusive as the credible interval includes zero.

  • diff_0_3: Estimate = -0.076 [-0.191, 0.033], Posterior Probability = 91.9%; Moderate evidence suggests a slight decrease in pitch_medianPost at percProm3 compared to percProm0, though it remains inconclusive.

  • diff_1_2: Estimate = -0.081 [-0.113, -0.05], Posterior Probability = 100%; Strong evidence of a decrease in pitch_medianPost from percProm1 to percProm2, as the credible interval lies entirely below zero.

  • diff_1_3: Estimate = -0.078 [-0.119, -0.036], Posterior Probability = 100%; Strong evidence that pitch_medianPost at percProm1 is lower than at percProm3, with a negative estimate and high posterior probability.

  • diff_2_3: Estimate = 0.004 [-0.031, 0.038], Posterior Probability = 58.1%; No strong evidence for a difference between percProm2 and percProm3.

Overall Implications:

  • Trend: There is strong evidence of a decreasing trend in pitch_medianPost from percProm1 to both percProm2 and percProm3. While the comparisons involving percProm0 are inconclusive, these findings suggest that as perceived prominence increases, pitch_medianPost generally decreases across most levels.
4.2.10.3.3 Plot fixef

Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.

4.2.10.4 Group-level effects

First, get a general overview.

  • percProm0: SD = 0.133 (95% CrI: 0.023 to 0.267), indicating minor variability in pitch_medianPost.

  • percProm1: SD = 0.092 (95% CrI: 0.065 to 0.125), suggesting lower variability in response at percProm1.

  • percProm2: SD = 0.079 (95% CrI: 0.059 to 0.105), indicating a further decrease in variability compared to percProm1.

  • percProm3: SD = 0.083 (95% CrI: 0.050 to 0.123), showing slightly more variability than percProm2.

Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.

First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.

This concludes the analysis.

5 Session info

sessionInfo()
## R version 4.4.1 (2024-06-14 ucrt)
## Platform: x86_64-w64-mingw32/x64
## Running under: Windows 10 x64 (build 19045)
## 
## Matrix products: default
## 
## 
## locale:
## [1] LC_COLLATE=German_Germany.utf8  LC_CTYPE=German_Germany.utf8   
## [3] LC_MONETARY=German_Germany.utf8 LC_NUMERIC=C                   
## [5] LC_TIME=German_Germany.utf8    
## 
## time zone: Europe/Berlin
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] distributional_0.5.0 gganimate_1.0.9      RColorBrewer_1.1-3  
##  [4] ggrepel_0.9.6        rstan_2.32.6         StanHeaders_2.32.10 
##  [7] cowplot_1.1.3        tidybayes_3.0.7      ggdist_3.3.2        
## [10] modelr_0.1.11        magrittr_2.0.3       posterior_1.6.0     
## [13] emmeans_1.10.5       cmdstanr_0.8.1       brms_2.22.0         
## [16] Rcpp_1.0.13          corrplot_0.94        gridExtra_2.3       
## [19] ggpubr_0.6.0         ggforce_0.4.2        data.table_1.16.0   
## [22] lubridate_1.9.3      forcats_1.0.0        dplyr_1.1.4         
## [25] purrr_1.0.2          readr_2.1.5          tidyr_1.3.1         
## [28] ggplot2_3.5.1        tidyverse_2.0.0      stringr_1.5.1       
## [31] tibble_3.2.1        
## 
## loaded via a namespace (and not attached):
##  [1] inline_0.3.19        rlang_1.1.4          matrixStats_1.4.1   
##  [4] compiler_4.4.1       mgcv_1.9-1           loo_2.8.0           
##  [7] systemfonts_1.1.0    reshape2_1.4.4       vctrs_0.6.5         
## [10] crayon_1.5.3         pkgconfig_2.0.3      arrayhelpers_1.1-0  
## [13] fastmap_1.2.0        backports_1.5.0      labeling_0.4.3      
## [16] utf8_1.2.4           rmarkdown_2.28       tzdb_0.4.0          
## [19] ps_1.8.0             ragg_1.3.3           bit_4.5.0           
## [22] xfun_0.48            cachem_1.1.0         jsonlite_1.8.9      
## [25] progress_1.2.3       highr_0.11           tweenr_2.0.3        
## [28] prettyunits_1.2.0    broom_1.0.7          parallel_4.4.1      
## [31] R6_2.5.1             bslib_0.8.0          stringi_1.8.4       
## [34] car_3.1-3            jquerylib_0.1.4      estimability_1.5.1  
## [37] knitr_1.48           bayesplot_1.11.1     splines_4.4.1       
## [40] Matrix_1.7-0         timechange_0.3.0     tidyselect_1.2.1    
## [43] rstudioapi_0.16.0    abind_1.4-8          yaml_2.3.10         
## [46] codetools_0.2-20     processx_3.8.4       pkgbuild_1.4.4      
## [49] plyr_1.8.9           lattice_0.22-6       withr_3.0.1         
## [52] bridgesampling_1.1-2 coda_0.19-4.1        evaluate_1.0.0      
## [55] RcppParallel_5.1.9   polyclip_1.10-7      pillar_1.9.0        
## [58] carData_3.0-5        tensorA_0.36.2.1     checkmate_2.3.2     
## [61] stats4_4.4.1         generics_0.1.3       vroom_1.6.5         
## [64] hms_1.1.3            rstantools_2.4.0     munsell_0.5.1       
## [67] scales_1.3.0         xtable_1.8-4         glue_1.8.0          
## [70] tools_4.4.1          ggsignif_0.6.4       mvtnorm_1.3-1       
## [73] grid_4.4.1           QuickJSR_1.4.0       colorspace_2.1-1    
## [76] nlme_3.1-164         Formula_1.2-5        cli_3.6.3           
## [79] textshaping_0.4.0    fansi_1.0.6          svUnit_1.0.6        
## [82] Brobdingnag_1.2-9    gtable_0.3.5         rstatix_0.7.2       
## [85] sass_0.4.9           digest_0.6.37        farver_2.1.2        
## [88] htmltools_0.5.8.1    lifecycle_1.0.4      bit64_4.5.2         
## [91] MASS_7.3-60.2